39 HFOPDE, chapter 2.2.5

 39.1 problem number 1
 39.2 problem number 2
 39.3 problem number 3
 39.4 problem number 4
 39.5 problem number 5
 39.6 problem number 6
 39.7 problem number 7
 39.8 problem number 8
 39.9 problem number 9
 39.10 problem number 10
 39.11 problem number 11
 39.12 problem number 12
 39.13 problem number 13
 39.14 problem number 14
 39.15 problem number 15
 39.16 problem number 16
 39.17 problem number 17
 39.18 problem number 18
 39.19 problem number 19
 39.20 problem number 20
 39.21 problem number 21
 39.22 problem number 22
 39.23 problem number 23
 39.24 problem number 24
 39.25 problem number 25
 39.26 problem number 26
 39.27 problem number 27
 39.28 problem number 28
 39.29 problem number 29
 39.30 problem number 30
 39.31 problem number 31
 39.32 problem number 32
 39.33 problem number 33
 39.34 problem number 34
 39.35 problem number 35
 39.36 problem number 36
 39.37 problem number 37
 39.38 problem number 38
 39.39 problem number 39
 39.40 problem number 40
 39.41 problem number 41
 39.42 problem number 42
 39.43 problem number 43
 39.44 problem number 44
 39.45 problem number 45
 39.46 problem number 46
 39.47 problem number 47
 39.48 problem number 48
 39.49 problem number 49
 39.50 problem number 50
 39.51 problem number 51
 39.52 problem number 52
 39.53 problem number 53
 39.54 problem number 54
 39.55 problem number 55
 39.56 problem number 56

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39.1 problem number 1

problem number 270

Added January 2, 2019.

Problem 2.2.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a y + b x^k \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (a^{-k-1} e^{-a x} \left (b e^{a x} \text{Gamma}(k+1,a x)+y a^{k+1}\right )\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{{{\rm e}^{-ax}} \left ( -{x}^{k} \left ( ax \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,ax \right ){{\rm e}^{1/2\,ax}}b+aky+ay \right ) }{a \left ( k+1 \right ) }} \right ) \]

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39.2 problem number 2

problem number 271

Added January 2, 2019.

Problem 2.2.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a x^k y+b x^n \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{a^{-\frac{n}{k+1}-\frac{1}{k+1}} e^{-\frac{a x^{k+1}}{k+1}} \left (b (k+1)^{\frac{n}{k+1}+\frac{1}{k+1}} e^{\frac{a x^{k+1}}{k+1}} \text{Gamma}\left (\frac{n}{k+1}+\frac{1}{k+1},\frac{a x^{k+1}}{k+1}\right )+k y a^{\frac{n}{k+1}+\frac{1}{k+1}}+y a^{\frac{n}{k+1}+\frac{1}{k+1}}\right )}{k+1}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{1}{a \left ( 2\,{k}^{2}n+3\,k{n}^{2}+{n}^{3}+2\,{k}^{2}+10\,kn+6\,{n}^{2}+7\,k+11\,n+6 \right ) } \left ( -6\,{{\rm e}^{-{\frac{{x}^{k+1}a}{k+1}}}}ya+4\,{{\rm e}^{-1/2\,{\frac{{x}^{k+1}a}{k+1}}}}{x}^{-k+n} \left ({\frac{{x}^{k+1}a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}} \WhittakerM \left ( 1/2\,{\frac{k+n+2}{k+1}},1/2\,{\frac{2\,k+n+3}{k+1}},{\frac{{x}^{k+1}a}{k+1}} \right ) b+{{\rm e}^{-1/2\,{\frac{{x}^{k+1}a}{k+1}}}}{x}^{-k+n} \left ({\frac{{x}^{k+1}a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}} \WhittakerM \left ( 1/2\,{\frac{k+n+2}{k+1}},1/2\,{\frac{2\,k+n+3}{k+1}},{\frac{{x}^{k+1}a}{k+1}} \right ) b{n}^{2}+5\,{{\rm e}^{-1/2\,{\frac{{x}^{k+1}a}{k+1}}}}{x}^{-k+n} \left ({\frac{{x}^{k+1}a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}} \WhittakerM \left ( -1/2\,{\frac{k-n}{k+1}},1/2\,{\frac{2\,k+n+3}{k+1}},{\frac{{x}^{k+1}a}{k+1}} \right ) bk+{{\rm e}^{-1/2\,{\frac{{x}^{k+1}a}{k+1}}}}{x}^{-k+n} \left ({\frac{{x}^{k+1}a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}} \WhittakerM \left ( -1/2\,{\frac{k-n}{k+1}},1/2\,{\frac{2\,k+n+3}{k+1}},{\frac{{x}^{k+1}a}{k+1}} \right ) bn+8\,{{\rm e}^{-1/2\,{\frac{{x}^{k+1}a}{k+1}}}}{x}^{-k+n} \left ({\frac{{x}^{k+1}a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}} \WhittakerM \left ( 1/2\,{\frac{k+n+2}{k+1}},1/2\,{\frac{2\,k+n+3}{k+1}},{\frac{{x}^{k+1}a}{k+1}} \right ) bk+4\,{{\rm e}^{-1/2\,{\frac{{x}^{k+1}a}{k+1}}}}{x}^{-k+n} \left ({\frac{{x}^{k+1}a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}} \WhittakerM \left ( 1/2\,{\frac{k+n+2}{k+1}},1/2\,{\frac{2\,k+n+3}{k+1}},{\frac{{x}^{k+1}a}{k+1}} \right ) bn+4\,{{\rm e}^{-1/2\,{\frac{{x}^{k+1}a}{k+1}}}}{x}^{-k+n} \left ({\frac{{x}^{k+1}a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}} \WhittakerM \left ( -1/2\,{\frac{k-n}{k+1}},1/2\,{\frac{2\,k+n+3}{k+1}},{\frac{{x}^{k+1}a}{k+1}} \right ) b{k}^{2}+{{\rm e}^{-1/2\,{\frac{{x}^{k+1}a}{k+1}}}}{x}^{-k+n} \left ({\frac{{x}^{k+1}a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}} \WhittakerM \left ( -1/2\,{\frac{k-n}{k+1}},1/2\,{\frac{2\,k+n+3}{k+1}},{\frac{{x}^{k+1}a}{k+1}} \right ) b{k}^{3}+{{\rm e}^{-1/2\,{\frac{{x}^{k+1}a}{k+1}}}}{x}^{-k+n} \left ({\frac{{x}^{k+1}a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}} \WhittakerM \left ( 1/2\,{\frac{k+n+2}{k+1}},1/2\,{\frac{2\,k+n+3}{k+1}},{\frac{{x}^{k+1}a}{k+1}} \right ) b{k}^{3}+5\,{{\rm e}^{-1/2\,{\frac{{x}^{k+1}a}{k+1}}}}{x}^{-k+n} \left ({\frac{{x}^{k+1}a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}} \WhittakerM \left ( 1/2\,{\frac{k+n+2}{k+1}},1/2\,{\frac{2\,k+n+3}{k+1}},{\frac{{x}^{k+1}a}{k+1}} \right ) b{k}^{2}+{{\rm e}^{-1/2\,{\frac{{x}^{k+1}a}{k+1}}}}{x}^{n+1} \left ({\frac{{x}^{k+1}a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}} \WhittakerM \left ( -1/2\,{\frac{k-n}{k+1}},1/2\,{\frac{2\,k+n+3}{k+1}},{\frac{{x}^{k+1}a}{k+1}} \right ) ab-3\,{{\rm e}^{-{\frac{{x}^{k+1}a}{k+1}}}}yak{n}^{2}-10\,{{\rm e}^{-{\frac{{x}^{k+1}a}{k+1}}}}yakn-2\,{{\rm e}^{-{\frac{{x}^{k+1}a}{k+1}}}}ya{k}^{2}n+2\,{{\rm e}^{-1/2\,{\frac{{x}^{k+1}a}{k+1}}}}{x}^{-k+n} \left ({\frac{{x}^{k+1}a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}} \WhittakerM \left ( -1/2\,{\frac{k-n}{k+1}},1/2\,{\frac{2\,k+n+3}{k+1}},{\frac{{x}^{k+1}a}{k+1}} \right ) b+2\,{{\rm e}^{-1/2\,{\frac{{x}^{k+1}a}{k+1}}}}{x}^{-k+n} \left ({\frac{{x}^{k+1}a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}} \WhittakerM \left ( -1/2\,{\frac{k-n}{k+1}},1/2\,{\frac{2\,k+n+3}{k+1}},{\frac{{x}^{k+1}a}{k+1}} \right ) bkn+6\,{{\rm e}^{-1/2\,{\frac{{x}^{k+1}a}{k+1}}}}{x}^{-k+n} \left ({\frac{{x}^{k+1}a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}} \WhittakerM \left ( 1/2\,{\frac{k+n+2}{k+1}},1/2\,{\frac{2\,k+n+3}{k+1}},{\frac{{x}^{k+1}a}{k+1}} \right ) bkn+{{\rm e}^{-1/2\,{\frac{{x}^{k+1}a}{k+1}}}}{x}^{n+1} \left ({\frac{{x}^{k+1}a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}} \WhittakerM \left ( -1/2\,{\frac{k-n}{k+1}},1/2\,{\frac{2\,k+n+3}{k+1}},{\frac{{x}^{k+1}a}{k+1}} \right ) ab{k}^{2}+2\,{{\rm e}^{-1/2\,{\frac{{x}^{k+1}a}{k+1}}}}{x}^{n+1} \left ({\frac{{x}^{k+1}a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}} \WhittakerM \left ( -1/2\,{\frac{k-n}{k+1}},1/2\,{\frac{2\,k+n+3}{k+1}},{\frac{{x}^{k+1}a}{k+1}} \right ) abk+{{\rm e}^{-1/2\,{\frac{{x}^{k+1}a}{k+1}}}}{x}^{-k+n} \left ({\frac{{x}^{k+1}a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}} \WhittakerM \left ( -1/2\,{\frac{k-n}{k+1}},1/2\,{\frac{2\,k+n+3}{k+1}},{\frac{{x}^{k+1}a}{k+1}} \right ) b{k}^{2}n+2\,{{\rm e}^{-1/2\,{\frac{{x}^{k+1}a}{k+1}}}}{x}^{-k+n} \left ({\frac{{x}^{k+1}a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}} \WhittakerM \left ( 1/2\,{\frac{k+n+2}{k+1}},1/2\,{\frac{2\,k+n+3}{k+1}},{\frac{{x}^{k+1}a}{k+1}} \right ) b{k}^{2}n+{{\rm e}^{-1/2\,{\frac{{x}^{k+1}a}{k+1}}}}{x}^{-k+n} \left ({\frac{{x}^{k+1}a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}} \WhittakerM \left ( 1/2\,{\frac{k+n+2}{k+1}},1/2\,{\frac{2\,k+n+3}{k+1}},{\frac{{x}^{k+1}a}{k+1}} \right ) bk{n}^{2}-2\,{{\rm e}^{-{\frac{{x}^{k+1}a}{k+1}}}}ya{k}^{2}-6\,{{\rm e}^{-{\frac{{x}^{k+1}a}{k+1}}}}ya{n}^{2}-7\,{{\rm e}^{-{\frac{{x}^{k+1}a}{k+1}}}}yak-11\,{{\rm e}^{-{\frac{{x}^{k+1}a}{k+1}}}}yan-{{\rm e}^{-{\frac{{x}^{k+1}a}{k+1}}}}ya{n}^{3} \right ) } \right ) \]

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39.3 problem number 3

problem number 272

Added January 2, 2019.

Problem 2.2.5.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a y^2+b x^n \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{2 \left (-a x y \text{BesselJ}\left (\frac{1}{n+2},\frac{2 \sqrt{a} \sqrt{b} x^{\frac{n+2}{2}}}{n+2}\right )-\sqrt{a} \sqrt{b} x^{\frac{n}{2}+1} \text{BesselJ}\left (\frac{1}{n+2}-1,\frac{2 \sqrt{a} \sqrt{b} x^{\frac{n+2}{2}}}{n+2}\right )\right )}{-\sqrt{a} \sqrt{b} x^{\frac{n}{2}+1} \text{BesselJ}\left (\frac{n+1}{n+2},\frac{2 \sqrt{a} \sqrt{b} x^{\frac{n}{2}+1}}{n+2}\right )+2 a x y \text{BesselJ}\left (-\frac{1}{n+2},\frac{2 \sqrt{a} \sqrt{b} x^{\frac{n+2}{2}}}{n+2}\right )+\text{BesselJ}\left (-\frac{1}{n+2},\frac{2 \sqrt{a} \sqrt{b} x^{\frac{n+2}{2}}}{n+2}\right )+\sqrt{a} \sqrt{b} x^{\frac{n}{2}+1} \text{BesselJ}\left (-\frac{n+3}{n+2},\frac{2 \sqrt{a} \sqrt{b} x^{\frac{n+2}{2}}}{n+2}\right )}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({1 \left ( y\BesselY \left ( \left ( n+2 \right ) ^{-1},2\,{\frac{\sqrt{ab}{x}^{n/2}x}{n+2}} \right ) ax-\BesselY \left ({\frac{n+3}{n+2}},2\,{\frac{\sqrt{ab}{x}^{n/2}x}{n+2}} \right ) \sqrt{ab}{x}^{n/2}x+\BesselY \left ( \left ( n+2 \right ) ^{-1},2\,{\frac{\sqrt{ab}{x}^{n/2}x}{n+2}} \right ) \right ) \left ( \BesselJ \left ({\frac{n+3}{n+2}},2\,{\frac{\sqrt{ab}{x}^{n/2}x}{n+2}} \right ) \sqrt{ab}{x}^{n/2}x-y\BesselJ \left ( \left ( n+2 \right ) ^{-1},2\,{\frac{\sqrt{ab}{x}^{n/2}x}{n+2}} \right ) ax-\BesselJ \left ( \left ( n+2 \right ) ^{-1},2\,{\frac{\sqrt{ab}{x}^{n/2}x}{n+2}} \right ) \right ) ^{-1}} \right ) \]

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39.4 problem number 4

problem number 273

Added January 2, 2019.

Problem 2.2.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( y^2+a n x^{n-1} -a^2 x^{2 n} \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (-a^2 x^{2 n}+a n x^{n-1}+y^2\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -2\,{({x}^{5/2\,n+2}a-{x}^{3/2\,n+2}y){{\rm e}^{-{\frac{{x}^{n+1}a}{n+1}}}} \left ( -a{n}^{2}{x}^{n+1} \WhittakerM \left ( -1/2\,{\frac{n}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ) +2\, \WhittakerM \left ( -1/2\,{\frac{n}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ){x}^{2\,n+2}{a}^{2}n-3\,a{n}^{2}{x}^{n+1} \WhittakerM \left ( 1/2\,{\frac{n+2}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ) -10\,an{x}^{n+1} \WhittakerM \left ( 1/2\,{\frac{n+2}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ) -3\,an{x}^{n+1} \WhittakerM \left ( -1/2\,{\frac{n}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ) -2\,ay \WhittakerM \left ( -1/2\,{\frac{n}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ){x}^{n+2}+{n}^{2}yx \WhittakerM \left ( 1/2\,{\frac{n+2}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ) +{n}^{2}yx \WhittakerM \left ( -1/2\,{\frac{n}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ) +4\,nyx \WhittakerM \left ( 1/2\,{\frac{n+2}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ) +3\,nyx \WhittakerM \left ( -1/2\,{\frac{n}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ) -2\,any \WhittakerM \left ( -1/2\,{\frac{n}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ){x}^{n+2}+12\, \left ( -2\,{\frac{{x}^{n+1}a}{n+1}} \right ) ^{1/2\,{\frac{3\,n+4}{n+1}}}{{\rm e}^{{\frac{{x}^{n+1}a}{n+1}}}}-2\, \WhittakerM \left ( -1/2\,{\frac{n}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ) -4\, \WhittakerM \left ( 1/2\,{\frac{n+2}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ) +2\,{n}^{3} \left ( -2\,{\frac{{x}^{n+1}a}{n+1}} \right ) ^{1/2\,{\frac{3\,n+4}{n+1}}}{{\rm e}^{{\frac{{x}^{n+1}a}{n+1}}}}+11\,{n}^{2} \left ( -2\,{\frac{{x}^{n+1}a}{n+1}} \right ) ^{1/2\,{\frac{3\,n+4}{n+1}}}{{\rm e}^{{\frac{{x}^{n+1}a}{n+1}}}}+20\,n \left ( -2\,{\frac{{x}^{n+1}a}{n+1}} \right ) ^{1/2\,{\frac{3\,n+4}{n+1}}}{{\rm e}^{{\frac{{x}^{n+1}a}{n+1}}}}-8\,a{x}^{n+1} \WhittakerM \left ( 1/2\,{\frac{n+2}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ) -2\,a{x}^{n+1} \WhittakerM \left ( -1/2\,{\frac{n}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ) +2\, \WhittakerM \left ( -1/2\,{\frac{n}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ){x}^{2\,n+2}{a}^{2}+4\,yx \WhittakerM \left ( 1/2\,{\frac{n+2}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ) +2\,yx \WhittakerM \left ( -1/2\,{\frac{n}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ) -5\, \WhittakerM \left ( -1/2\,{\frac{n}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ) n-{n}^{3} \WhittakerM \left ( 1/2\,{\frac{n+2}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ) -{n}^{3} \WhittakerM \left ( -1/2\,{\frac{n}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ) -5\,{n}^{2} \WhittakerM \left ( 1/2\,{\frac{n+2}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ) -4\,{n}^{2} \WhittakerM \left ( -1/2\,{\frac{n}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ) -8\,n \WhittakerM \left ( 1/2\,{\frac{n+2}{n+1}},1/2\,{\frac{2\,n+3}{n+1}},-2\,{\frac{{x}^{n+1}a}{n+1}} \right ) \right ) ^{-1}} \right ) \]

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39.5 problem number 5

problem number 274

Added January 2, 2019.

Problem 2.2.5.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( y^2 + a x^n y + a x^{n-1} \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (-\frac{(n+1)^{-\frac{1}{n+1}-1} \left ((-1)^{\frac{1}{n+1}+1} x a^{\frac{1}{n+1}} \text{Gamma}\left (\frac{1}{-n-1},-\frac{a x^{n+1}}{n+1}\right )+(-1)^{\frac{1}{n+1}+1} x^2 y a^{\frac{1}{n+1}} \text{Gamma}\left (\frac{1}{-n-1},-\frac{a x^{n+1}}{n+1}\right )+(n+1)^{\frac{1}{n+1}} e^{\frac{a x^{n+1}}{n+1}}+n (n+1)^{\frac{1}{n+1}} e^{\frac{a x^{n+1}}{n+1}}\right )}{x (x y+1)}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{nxa \left ( 2\,nyx+yx+2\,n+1 \right ) } \left ({{\rm e}^{1/2\,{\frac{{x}^{n+1}a}{n+1}}}}y \left ( -{\frac{{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac{n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac{n+2}{n+1}},1/2\,{\frac{1+2\,n}{n+1}},-{\frac{{x}^{n+1}a}{n+1}} \right ){x}^{-n}{n}^{3}+{{\rm e}^{1/2\,{\frac{{x}^{n+1}a}{n+1}}}}y \left ( -{\frac{{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac{n}{n+1}}} \WhittakerM \left ( 1/2\,{\frac{n}{n+1}},1/2\,{\frac{1+2\,n}{n+1}},-{\frac{{x}^{n+1}a}{n+1}} \right ){x}^{-n}{n}^{3}+2\,{{\rm e}^{1/2\,{\frac{{x}^{n+1}a}{n+1}}}}y \left ( -{\frac{{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac{n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac{n+2}{n+1}},1/2\,{\frac{1+2\,n}{n+1}},-{\frac{{x}^{n+1}a}{n+1}} \right ){x}^{-n}{n}^{2}-{{\rm e}^{1/2\,{\frac{{x}^{n+1}a}{n+1}}}}y \left ( -{\frac{{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac{n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac{n+2}{n+1}},1/2\,{\frac{1+2\,n}{n+1}},-{\frac{{x}^{n+1}a}{n+1}} \right ) a{n}^{2}x+{{\rm e}^{1/2\,{\frac{{x}^{n+1}a}{n+1}}}}y \left ( -{\frac{{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac{n}{n+1}}} \WhittakerM \left ( 1/2\,{\frac{n}{n+1}},1/2\,{\frac{1+2\,n}{n+1}},-{\frac{{x}^{n+1}a}{n+1}} \right ){x}^{-n}{n}^{2}+{{\rm e}^{1/2\,{\frac{{x}^{n+1}a}{n+1}}}} \left ( -{\frac{{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac{n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac{n+2}{n+1}},1/2\,{\frac{1+2\,n}{n+1}},-{\frac{{x}^{n+1}a}{n+1}} \right ){x}^{-n-1}{n}^{3}+{{\rm e}^{1/2\,{\frac{{x}^{n+1}a}{n+1}}}} \left ( -{\frac{{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac{n}{n+1}}} \WhittakerM \left ( 1/2\,{\frac{n}{n+1}},1/2\,{\frac{1+2\,n}{n+1}},-{\frac{{x}^{n+1}a}{n+1}} \right ){x}^{-n-1}{n}^{3}+{{\rm e}^{1/2\,{\frac{{x}^{n+1}a}{n+1}}}}y \left ( -{\frac{{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac{n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac{n+2}{n+1}},1/2\,{\frac{1+2\,n}{n+1}},-{\frac{{x}^{n+1}a}{n+1}} \right ){x}^{-n}n-2\,{{\rm e}^{1/2\,{\frac{{x}^{n+1}a}{n+1}}}}y \left ( -{\frac{{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac{n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac{n+2}{n+1}},1/2\,{\frac{1+2\,n}{n+1}},-{\frac{{x}^{n+1}a}{n+1}} \right ) anx+2\,{{\rm e}^{1/2\,{\frac{{x}^{n+1}a}{n+1}}}} \left ( -{\frac{{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac{n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac{n+2}{n+1}},1/2\,{\frac{1+2\,n}{n+1}},-{\frac{{x}^{n+1}a}{n+1}} \right ){x}^{-n-1}{n}^{2}+{{\rm e}^{1/2\,{\frac{{x}^{n+1}a}{n+1}}}} \left ( -{\frac{{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac{n}{n+1}}} \WhittakerM \left ( 1/2\,{\frac{n}{n+1}},1/2\,{\frac{1+2\,n}{n+1}},-{\frac{{x}^{n+1}a}{n+1}} \right ){x}^{-n-1}{n}^{2}-{{\rm e}^{1/2\,{\frac{{x}^{n+1}a}{n+1}}}}y \left ( -{\frac{{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac{n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac{n+2}{n+1}},1/2\,{\frac{1+2\,n}{n+1}},-{\frac{{x}^{n+1}a}{n+1}} \right ) ax+{{\rm e}^{1/2\,{\frac{{x}^{n+1}a}{n+1}}}} \left ( -{\frac{{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac{n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac{n+2}{n+1}},1/2\,{\frac{1+2\,n}{n+1}},-{\frac{{x}^{n+1}a}{n+1}} \right ){x}^{-n-1}n-{{\rm e}^{1/2\,{\frac{{x}^{n+1}a}{n+1}}}} \left ( -{\frac{{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac{n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac{n+2}{n+1}},1/2\,{\frac{1+2\,n}{n+1}},-{\frac{{x}^{n+1}a}{n+1}} \right ) a{n}^{2}-2\,{{\rm e}^{1/2\,{\frac{{x}^{n+1}a}{n+1}}}} \left ( -{\frac{{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac{n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac{n+2}{n+1}},1/2\,{\frac{1+2\,n}{n+1}},-{\frac{{x}^{n+1}a}{n+1}} \right ) an+2\,{{\rm e}^{{\frac{{x}^{n+1}a}{n+1}}}}a{n}^{2}-{{\rm e}^{1/2\,{\frac{{x}^{n+1}a}{n+1}}}} \left ( -{\frac{{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac{n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac{n+2}{n+1}},1/2\,{\frac{1+2\,n}{n+1}},-{\frac{{x}^{n+1}a}{n+1}} \right ) a+{{\rm e}^{{\frac{{x}^{n+1}a}{n+1}}}}an \right ) } \right ) \]

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39.6 problem number 6

problem number 275

Added January 2, 2019.

Problem 2.2.5.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( y^2+a x^n y-a b x^n -b^2 \right ) w_y = 0 \]

Mathematica

\[ \text{Timed out} \] Timed out

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{1}{-b+y} \left ( \int \!{{\rm e}^{{\frac{x \left ({x}^{n}a+2\,bn+2\,b \right ) }{n+1}}}}\,{\rm d}xy-\int \!{{\rm e}^{{\frac{x \left ({x}^{n}a+2\,bn+2\,b \right ) }{n+1}}}}\,{\rm d}xb+{{\rm e}^{{\frac{x \left ({x}^{n}a+2\,bn+2\,b \right ) }{n+1}}}} \right ) } \right ) \]

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39.7 problem number 7

problem number 276

Added January 2, 2019.

Problem 2.2.5.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a x^n y^2 + b x^{-n-2} \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{-2 a y x^{\frac{1}{2} \sqrt{a} \sqrt{b} \left (\frac{\sqrt{-4 a b+n^2+2 n+1}}{\sqrt{a} \sqrt{b}}-\frac{-n-1}{\sqrt{a} \sqrt{b}}\right )+n+1}-n x^{\frac{1}{2} \sqrt{a} \sqrt{b} \left (\frac{\sqrt{-4 a b+n^2+2 n+1}}{\sqrt{a} \sqrt{b}}-\frac{-n-1}{\sqrt{a} \sqrt{b}}\right )}-\sqrt{-4 a b+n^2+2 n+1} x^{\frac{1}{2} \sqrt{a} \sqrt{b} \left (\frac{\sqrt{-4 a b+n^2+2 n+1}}{\sqrt{a} \sqrt{b}}-\frac{-n-1}{\sqrt{a} \sqrt{b}}\right )}-x^{\frac{1}{2} \sqrt{a} \sqrt{b} \left (\frac{\sqrt{-4 a b+n^2+2 n+1}}{\sqrt{a} \sqrt{b}}-\frac{-n-1}{\sqrt{a} \sqrt{b}}\right )}}{2 a y x^{\frac{1}{2} \sqrt{a} \sqrt{b} \left (-\frac{\sqrt{-4 a b+n^2+2 n+1}}{\sqrt{a} \sqrt{b}}-\frac{-n-1}{\sqrt{a} \sqrt{b}}\right )+n+1}+n x^{\frac{1}{2} \sqrt{a} \sqrt{b} \left (-\frac{\sqrt{-4 a b+n^2+2 n+1}}{\sqrt{a} \sqrt{b}}-\frac{-n-1}{\sqrt{a} \sqrt{b}}\right )}-\sqrt{-4 a b+n^2+2 n+1} x^{\frac{1}{2} \sqrt{a} \sqrt{b} \left (-\frac{\sqrt{-4 a b+n^2+2 n+1}}{\sqrt{a} \sqrt{b}}-\frac{-n-1}{\sqrt{a} \sqrt{b}}\right )}+x^{\frac{1}{2} \sqrt{a} \sqrt{b} \left (-\frac{\sqrt{-4 a b+n^2+2 n+1}}{\sqrt{a} \sqrt{b}}-\frac{-n-1}{\sqrt{a} \sqrt{b}}\right )}}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{\sqrt{4\,ab-{n}^{2}-2\,n-1}} \left ( \ln \left ( x \right ) \sqrt{4\,ab-{n}^{2}-2\,n-1}-2\,\arctan \left ({\frac{2\,a{x}^{n}yx+n+1}{\sqrt{4\,ab-{n}^{2}-2\,n-1}}} \right ) \right ) } \right ) \]

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39.8 problem number 8

problem number 277

Added January 2, 2019.

Problem 2.2.5.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a x^n y^2 + b m x^{m-1} -a b^2 x^{n+2 m} \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (-a b^2 x^{2 m+n}+a y^2 x^n+b m x^{m-1}\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( 2\,{a \left ( -{x}^{5/2\,m+2\,n+2}b+{x}^{3/2\,m+2\,n+2}y \right ){{\rm e}^{-{\frac{{x}^{m+n+1}ab}{m+n+1}}}} \left ( 12\,{n}^{3} \left ( -2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) ^{1/2\,{\frac{3\,m+4\,n+4}{m+n+1}}}{{\rm e}^{{\frac{{x}^{m+n+1}ab}{m+n+1}}}}+11\,{m}^{2} \left ( -2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) ^{1/2\,{\frac{3\,m+4\,n+4}{m+n+1}}}{{\rm e}^{{\frac{{x}^{m+n+1}ab}{m+n+1}}}}+20\,m{n}^{2} \left ( -2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) ^{1/2\,{\frac{3\,m+4\,n+4}{m+n+1}}}{{\rm e}^{{\frac{{x}^{m+n+1}ab}{m+n+1}}}}+40\,mn \left ( -2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) ^{1/2\,{\frac{3\,m+4\,n+4}{m+n+1}}}{{\rm e}^{{\frac{{x}^{m+n+1}ab}{m+n+1}}}}+11\,{m}^{2}n \left ( -2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) ^{1/2\,{\frac{3\,m+4\,n+4}{m+n+1}}}{{\rm e}^{{\frac{{x}^{m+n+1}ab}{m+n+1}}}}-8\,ab{x}^{m+n+1} \WhittakerM \left ( 1/2\,{\frac{m+2\,n+2}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -2\,ab{x}^{m+n+1} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) +2\,{x}^{2\,n+2\,m+2} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ){a}^{2}{b}^{2}+4\,ay{x}^{n+1} \WhittakerM \left ( 1/2\,{\frac{m+2\,n+2}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) +2\,ay{x}^{n+1} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -4\, \WhittakerM \left ( 1/2\,{\frac{m+2\,n+2}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -2\, \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) +12\, \left ( -2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) ^{1/2\,{\frac{3\,m+4\,n+4}{m+n+1}}}{{\rm e}^{{\frac{{x}^{m+n+1}ab}{m+n+1}}}}+36\,{n}^{2} \left ( -2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) ^{1/2\,{\frac{3\,m+4\,n+4}{m+n+1}}}{{\rm e}^{{\frac{{x}^{m+n+1}ab}{m+n+1}}}}+20\,m \left ( -2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) ^{1/2\,{\frac{3\,m+4\,n+4}{m+n+1}}}{{\rm e}^{{\frac{{x}^{m+n+1}ab}{m+n+1}}}}+2\,{m}^{3} \left ( -2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) ^{1/2\,{\frac{3\,m+4\,n+4}{m+n+1}}}{{\rm e}^{{\frac{{x}^{m+n+1}ab}{m+n+1}}}}+36\,n \left ( -2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) ^{1/2\,{\frac{3\,m+4\,n+4}{m+n+1}}}{{\rm e}^{{\frac{{x}^{m+n+1}ab}{m+n+1}}}}-10\,abmn{x}^{m+n+1} \WhittakerM \left ( 1/2\,{\frac{m+2\,n+2}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -3\,abmn{x}^{m+n+1} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) +4\,amyn{x}^{n+1} \WhittakerM \left ( 1/2\,{\frac{m+2\,n+2}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) +3\,amyn{x}^{n+1} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -2\,y{x}^{m+2\,n+2} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ){a}^{2}bm-2\,y{x}^{m+2\,n+2} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ){a}^{2}bn+2\,{x}^{2\,n+2\,m+2} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ){a}^{2}{b}^{2}m-ab{m}^{2}{x}^{m+n+1} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) +2\,{x}^{2\,n+2\,m+2} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ){a}^{2}{b}^{2}n-3\,ab{m}^{2}{x}^{m+n+1} \WhittakerM \left ( 1/2\,{\frac{m+2\,n+2}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -10\,abm{x}^{m+n+1} \WhittakerM \left ( 1/2\,{\frac{m+2\,n+2}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -3\,abm{x}^{m+n+1} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -8\,ab{n}^{2}{x}^{m+n+1} \WhittakerM \left ( 1/2\,{\frac{m+2\,n+2}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -2\,ab{n}^{2}{x}^{m+n+1} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -16\,abn{x}^{m+n+1} \WhittakerM \left ( 1/2\,{\frac{m+2\,n+2}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -4\,abn{x}^{m+n+1} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) +ay{m}^{2}{x}^{n+1} \WhittakerM \left ( 1/2\,{\frac{m+2\,n+2}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) +ay{m}^{2}{x}^{n+1} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -2\,y{x}^{m+2\,n+2} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ){a}^{2}b+4\,amy{x}^{n+1} \WhittakerM \left ( 1/2\,{\frac{m+2\,n+2}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) +3\,amy{x}^{n+1} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) +4\,ay{n}^{2}{x}^{n+1} \WhittakerM \left ( 1/2\,{\frac{m+2\,n+2}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) +2\,ay{n}^{2}{x}^{n+1} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) +8\,ayn{x}^{n+1} \WhittakerM \left ( 1/2\,{\frac{m+2\,n+2}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) +4\,ayn{x}^{n+1} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -{m}^{3} \WhittakerM \left ( 1/2\,{\frac{m+2\,n+2}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -{m}^{3} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -12\,n \WhittakerM \left ( 1/2\,{\frac{m+2\,n+2}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -5\,{m}^{2} \WhittakerM \left ( 1/2\,{\frac{m+2\,n+2}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -4\,{m}^{2} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -12\,{n}^{2} \WhittakerM \left ( 1/2\,{\frac{m+2\,n+2}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -6\,{n}^{2} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -8\,m \WhittakerM \left ( 1/2\,{\frac{m+2\,n+2}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -4\,{n}^{3} \WhittakerM \left ( 1/2\,{\frac{m+2\,n+2}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -2\,{n}^{3} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -5\, \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) m-6\, \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) n-5\,{m}^{2}n \WhittakerM \left ( 1/2\,{\frac{m+2\,n+2}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -5\,m{n}^{2} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -8\,m{n}^{2} \WhittakerM \left ( 1/2\,{\frac{m+2\,n+2}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -4\,{m}^{2} \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) n-16\,mn \WhittakerM \left ( 1/2\,{\frac{m+2\,n+2}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) -10\,m \WhittakerM \left ( -1/2\,{\frac{m}{m+n+1}},1/2\,{\frac{2\,m+3\,n+3}{m+n+1}},-2\,{\frac{{x}^{m+n+1}ab}{m+n+1}} \right ) n \right ) ^{-1}} \right ) \]

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39.9 problem number 9

problem number 278

Added January 2, 2019.

Problem 2.2.5.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( (n+1)x^n y^2 - a x^{n+m+1} y + a x^m \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (-a y x^{m+n+1}+a x^m+(n+1) y^2 x^n\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{1 \left ( y{\mbox{$_1$F$_1$}({\frac{-n+m}{m+n+2}};\,{\frac{m+1}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{m}^{2}+2\,y{\mbox{$_1$F$_1$}({\frac{-n+m}{m+n+2}};\,{\frac{m+1}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{n}^{2}+4\,y{\mbox{$_1$F$_1$}({\frac{-n+m}{m+n+2}};\,{\frac{m+1}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}m+5\,y{\mbox{$_1$F$_1$}({\frac{-n+m}{m+n+2}};\,{\frac{m+1}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}n+y{\mbox{$_1$F$_1$}({\frac{-n+m}{m+n+2}};\,{\frac{m+1}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{m}^{2}n+2\,y{\mbox{$_1$F$_1$}({\frac{-n+m}{m+n+2}};\,{\frac{m+1}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}m{n}^{2}-3\,{\mbox{$_1$F$_1$}({\frac{-n+m}{m+n+2}};\,{\frac{m+1}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{m}ax+6\,y{\mbox{$_1$F$_1$}({\frac{-n+m}{m+n+2}};\,{\frac{m+1}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}mn-2\,{\mbox{$_1$F$_1$}({\frac{-n+m}{m+n+2}};\,{\frac{m+1}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{m}amnx+{\mbox{$_1$F$_1$}(2\,{\frac{m+1}{m+n+2}};\,{\frac{2\,m+3+n}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{m}amnx-{\mbox{$_1$F$_1$}({\frac{-n+m}{m+n+2}};\,{\frac{m+1}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{m}a{m}^{2}x+{\mbox{$_1$F$_1$}(2\,{\frac{m+1}{m+n+2}};\,{\frac{2\,m+3+n}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{m}a{m}^{2}x-2\,{\mbox{$_1$F$_1$}(2\,{\frac{m+1}{m+n+2}};\,{\frac{2\,m+3+n}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{m}a{n}^{2}x-4\,{\mbox{$_1$F$_1$}({\frac{-n+m}{m+n+2}};\,{\frac{m+1}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{m}amx-2\,{\mbox{$_1$F$_1$}({\frac{-n+m}{m+n+2}};\,{\frac{m+1}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{m}anx+3\,{\mbox{$_1$F$_1$}(2\,{\frac{m+1}{m+n+2}};\,{\frac{2\,m+3+n}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{m}amx-3\,{\mbox{$_1$F$_1$}(2\,{\frac{m+1}{m+n+2}};\,{\frac{2\,m+3+n}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{m}anx+3\,y{\mbox{$_1$F$_1$}({\frac{-n+m}{m+n+2}};\,{\frac{m+1}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})} \right ) \left ({\mbox{$_1$F$_1$}({\frac{m+1}{m+n+2}};\,{\frac{m+2\,n+3}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{m}^{2}+2\,{\mbox{$_1$F$_1$}({\frac{m+1}{m+n+2}};\,{\frac{m+2\,n+3}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{n}^{2}+4\,{\mbox{$_1$F$_1$}({\frac{m+1}{m+n+2}};\,{\frac{m+2\,n+3}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}m+5\,{\mbox{$_1$F$_1$}({\frac{m+1}{m+n+2}};\,{\frac{m+2\,n+3}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}n+{\mbox{$_1$F$_1$}({\frac{m+1}{m+n+2}};\,{\frac{m+2\,n+3}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{m}^{2}n+2\,{\mbox{$_1$F$_1$}({\frac{m+1}{m+n+2}};\,{\frac{m+2\,n+3}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}m{n}^{2}+6\,{\mbox{$_1$F$_1$}({\frac{m+1}{m+n+2}};\,{\frac{m+2\,n+3}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}mn+y{\mbox{$_1$F$_1$}({\frac{m+1}{m+n+2}};\,{\frac{m+2\,n+3}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}{m}^{2}nx+2\,y{\mbox{$_1$F$_1$}({\frac{m+1}{m+n+2}};\,{\frac{m+2\,n+3}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}m{n}^{2}x+6\,y{\mbox{$_1$F$_1$}({\frac{m+1}{m+n+2}};\,{\frac{m+2\,n+3}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}mnx-{\mbox{$_1$F$_1$}({\frac{m+1}{m+n+2}};\,{\frac{m+2\,n+3}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}{x}^{m}a{m}^{2}{x}^{2}+{\mbox{$_1$F$_1$}({\frac{2\,m+3+n}{m+n+2}};\,{\frac{2\,m+3\,n+5}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}{x}^{m}a{m}^{2}{x}^{2}-4\,{\mbox{$_1$F$_1$}({\frac{m+1}{m+n+2}};\,{\frac{m+2\,n+3}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}{x}^{m}am{x}^{2}-2\,{\mbox{$_1$F$_1$}({\frac{m+1}{m+n+2}};\,{\frac{m+2\,n+3}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}{x}^{m}an{x}^{2}+2\,{\mbox{$_1$F$_1$}({\frac{2\,m+3+n}{m+n+2}};\,{\frac{2\,m+3\,n+5}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}{x}^{m}am{x}^{2}+2\,y{\mbox{$_1$F$_1$}({\frac{m+1}{m+n+2}};\,{\frac{m+2\,n+3}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}{n}^{2}x+4\,y{\mbox{$_1$F$_1$}({\frac{m+1}{m+n+2}};\,{\frac{m+2\,n+3}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}mx+5\,y{\mbox{$_1$F$_1$}({\frac{m+1}{m+n+2}};\,{\frac{m+2\,n+3}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}nx+3\,y{\mbox{$_1$F$_1$}({\frac{m+1}{m+n+2}};\,{\frac{m+2\,n+3}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}x-2\,{\mbox{$_1$F$_1$}({\frac{m+1}{m+n+2}};\,{\frac{m+2\,n+3}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}{x}^{m}amn{x}^{2}-3\,{\mbox{$_1$F$_1$}({\frac{m+1}{m+n+2}};\,{\frac{m+2\,n+3}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}{x}^{m}a{x}^{2}+{x}^{m}{x}^{n}{x}^{2}a{\mbox{$_1$F$_1$}({\frac{2\,m+3+n}{m+n+2}};\,{\frac{2\,m+3\,n+5}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}+y{\mbox{$_1$F$_1$}({\frac{m+1}{m+n+2}};\,{\frac{m+2\,n+3}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}{m}^{2}x+3\,{\mbox{$_1$F$_1$}({\frac{m+1}{m+n+2}};\,{\frac{m+2\,n+3}{m+n+2}};\,{\frac{{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})} \right ) ^{-1}} \right ) \]

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39.10 problem number 10

problem number 279

Added January 2, 2019.

Problem 2.2.5.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a x^n y^2 + b x^m y+ b c x^m -a c^2 x^n \right ) w_y = 0 \]

Mathematica

\[ \text{Timed out} \] Timed out

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{1}{c+y} \left ( \int \!{{\rm e}^{{\frac{x \left ( -2\,{x}^{n}acm+{x}^{m}bn-2\,{x}^{n}ac+{x}^{m}b \right ) }{ \left ( m+1 \right ) \left ( n+1 \right ) }}}}{x}^{n}a\,{\rm d}xy+\int \!{{\rm e}^{{\frac{x \left ( -2\,{x}^{n}acm+{x}^{m}bn-2\,{x}^{n}ac+{x}^{m}b \right ) }{ \left ( m+1 \right ) \left ( n+1 \right ) }}}}{x}^{n}a\,{\rm d}xc+{{\rm e}^{-{\frac{x \left ( 2\,{x}^{n}acm+2\,{x}^{n}ac-{x}^{m}bn-{x}^{m}b \right ) }{ \left ( m+1 \right ) \left ( n+1 \right ) }}}} \right ) } \right ) \]

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39.11 problem number 11

problem number 280

Added January 2, 2019.

Problem 2.2.5.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a x^n y^2-a x^n (b x^m +c) y+ b m x^{m-1} \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (-a y x^n \left (b x^m+c\right )+a y^2 x^n+b m x^{m-1}\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ \text{ sol=() } \]

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39.12 problem number 12

problem number 281

Added January 2, 2019.

Problem 2.2.5.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x - \left (a n x^{n-1} y^2 - c x^m (a x^n+b) + c x^m \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(1,0)}(x,y)-w^{(0,1)}(x,y) \left (-c x^m \left (a x^n+b\right )+a n y^2 x^{n-1}+c x^m\right )=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ \text{ sol=() } \]

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39.13 problem number 13

problem number 282

Added January 2, 2019.

Problem 2.2.5.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a x^n y^2+b x^m y+ c k x^{k-1}-b c x^{m+k}-a c^2 x^{n+2 k} \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (-a c^2 x^{2 k+n}+a y^2 x^n-b c x^{k+m}+b y x^m+c k x^{k-1}\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ \text{ sol=() } \]

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39.14 problem number 14

problem number 283

Added January 2, 2019.

Problem 2.2.5.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a x^{2 n+1} y^3 + b x^{-n-2} \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (a y^3 x^{2 n+1}+b x^{-n-2}\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( \ln \left ( x \right ) -\sum _{{\it \_R}=\RootOf \left ( a{{\it \_Z}}^{3}+{\it \_Z}\, \left ( n+1 \right ) +b \right ) }{\frac{\ln \left ( y{x}^{n}x-{\it \_R} \right ) }{3\,{{\it \_R}}^{2}a+n+1}} \right ) \] Solution contains RootOf

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39.15 problem number 15

problem number 284

Added January 2, 2019.

Problem 2.2.5.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a x^n y^3 + 3 a b x^{n+m} y^2 - b m x^{m-1} - 2 a b^3 x^{n+3 m} \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{6^{-\frac{n}{2 m+n+1}-\frac{1}{2 m+n+1}} (2 m+n+1)^{-\frac{2 m}{2 m+n+1}} b^{-\frac{2 n}{2 m+n+1}-\frac{2}{2 m+n+1}} e^{-\frac{6 a b^2 x^{2 m+n+1}}{2 m+n+1}} \left (-2 y^2 a^{\frac{2 m}{2 m+n+1}} e^{\frac{6 a b^2 x^{2 m+n+1}}{2 m+n+1}} \text{Gamma}\left (\frac{n+1}{2 m+n+1},\frac{6 a b^2 x^{2 m+n+1}}{2 m+n+1}\right )-2 b^2 x^{2 m} a^{\frac{2 m}{2 m+n+1}} e^{\frac{6 a b^2 x^{2 m+n+1}}{2 m+n+1}} \text{Gamma}\left (\frac{n+1}{2 m+n+1},\frac{6 a b^2 x^{2 m+n+1}}{2 m+n+1}\right )-4 b y x^m a^{\frac{2 m}{2 m+n+1}} e^{\frac{6 a b^2 x^{2 m+n+1}}{2 m+n+1}} \text{Gamma}\left (\frac{n+1}{2 m+n+1},\frac{6 a b^2 x^{2 m+n+1}}{2 m+n+1}\right )+6^{\frac{n}{2 m+n+1}+\frac{1}{2 m+n+1}} (2 m+n+1)^{\frac{2 m}{2 m+n+1}} b^{\frac{2 n}{2 m+n+1}+\frac{2}{2 m+n+1}}\right )}{x^{2 m} b^{\frac{4 m}{2 m+n+1}+\frac{2 n}{2 m+n+1}+\frac{2}{2 m+n+1}}+2 b y x^m+y^2}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{{b}^{2} \left ( 3\,{y}^{2}+6\,b{x}^{m}y+14\,{x}^{m}bmy+4\,{y}^{2}{m}^{2}n+7\,{y}^{2}m{n}^{2}+3\,{y}^{2}{n}^{3}+4\,{y}^{2}{m}^{2}+9\,{y}^{2}{n}^{2}+7\,{y}^{2}m+9\,{y}^{2}n+4\,{x}^{2\,m}{b}^{2}{m}^{2}+9\,{x}^{2\,m}{b}^{2}{n}^{2}+7\,{x}^{2\,m}{b}^{2}m+9\,{x}^{2\,m}{b}^{2}n+3\,{x}^{2\,m}{b}^{2}{n}^{3}+8\,{m}^{2}{x}^{m}by+28\,y{x}^{m}bmn+8\,y{x}^{m}b{m}^{2}n+14\,y{x}^{m}bm{n}^{2}+4\,{x}^{2\,m}{b}^{2}{m}^{2}n+7\,{x}^{2\,m}{b}^{2}m{n}^{2}+14\,{x}^{2\,m}{b}^{2}mn+3\,{x}^{2\,m}{b}^{2}+6\,y{x}^{m}b{n}^{3}+18\,y{x}^{m}b{n}^{2}+18\,y{x}^{m}bn+14\,{y}^{2}mn \right ) } \left ( 21\,{{\rm e}^{-6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{9}^{-{\frac{m}{2\,m+n+1}}}{3}^{-{\frac{n+1}{2\,m+n+1}}}{b}^{2}m{n}^{2}+42\,{{\rm e}^{-6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{9}^{-{\frac{m}{2\,m+n+1}}}{3}^{-{\frac{n+1}{2\,m+n+1}}}{b}^{2}mn+4\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-2\,m}{y}^{2} \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){m}^{3}+4\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-2\,m}{y}^{2} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){m}^{3}+10\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){b}^{2}mn+3\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{2\,m+n+1} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) a{b}^{4}+5\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){b}^{2}m{n}^{2}+10\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}} \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){b}^{2}{m}^{2}n+8\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}} \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){b}^{2}m{n}^{2}+{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-2\,m}{y}^{2} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){n}^{3}+8\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-2\,m}{y}^{2} \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) m+6\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-2\,m}{y}^{2} \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) n+6\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-2\,m}{y}^{2} \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){n}^{2}+5\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-2\,m}{y}^{2} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) m+3\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-2\,m}{y}^{2} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) n+{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{2}^{{\frac{m}{2\,m+n+1}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-m}y \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) b+{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{2}^{{\frac{m}{2\,m+n+1}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-2\,m}{y}^{2} \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){n}^{3}+4\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-m}y \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) b+8\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-2\,m}{y}^{2} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){m}^{2}+3\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-2\,m}{y}^{2} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){n}^{2}+10\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-2\,m}{y}^{2} \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){m}^{2}+16\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}} \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){b}^{2}mn+8\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){b}^{2}{m}^{2}n+{{\rm e}^{-6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{9}^{{\frac{m+n+1}{2\,m+n+1}}}{3}^{-{\frac{n+1}{2\,m+n+1}}}{b}^{2}{n}^{3}+10\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-2\,m}{y}^{2} \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){m}^{2}n+8\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-2\,m}{y}^{2} \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) m{n}^{2}+12\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{2\,m+n+1} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) a{b}^{4}{m}^{2}+3\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{2\,m+n+1} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) a{b}^{4}{n}^{2}+12\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{2\,m+n+1} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) a{b}^{4}m+24\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{m+n+1} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) a{b}^{3}mny+12\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{n+1} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) a{b}^{2}mn{y}^{2}+{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{2}^{{\frac{m}{2\,m+n+1}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}} \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){b}^{2}+{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){b}^{2}+27\,{{\rm e}^{-6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{9}^{-{\frac{m}{2\,m+n+1}}}{3}^{-{\frac{n+1}{2\,m+n+1}}}{b}^{2}{n}^{2}+21\,{{\rm e}^{-6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{9}^{-{\frac{m}{2\,m+n+1}}}{3}^{-{\frac{n+1}{2\,m+n+1}}}{b}^{2}m+27\,{{\rm e}^{-6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{9}^{-{\frac{m}{2\,m+n+1}}}{3}^{-{\frac{n+1}{2\,m+n+1}}}{b}^{2}n+{{\rm e}^{-6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{9}^{{\frac{m+n+1}{2\,m+n+1}}}{3}^{-{\frac{n+1}{2\,m+n+1}}}{b}^{2}+12\,{{\rm e}^{-6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{9}^{-{\frac{m}{2\,m+n+1}}}{3}^{-{\frac{n+1}{2\,m+n+1}}}{b}^{2}{m}^{2}+4\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){b}^{2}{m}^{3}+6\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}} \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){b}^{2}{n}^{2}+8\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}} \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){b}^{2}m+6\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}} \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){b}^{2}n+{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){b}^{2}{n}^{3}+8\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){b}^{2}{m}^{2}+3\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){b}^{2}{n}^{2}+10\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}} \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){b}^{2}{m}^{2}+5\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){b}^{2}m+3\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){b}^{2}n+4\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}} \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){b}^{2}{m}^{3}+{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{2}^{{\frac{m}{2\,m+n+1}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}} \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){b}^{2}{n}^{3}+3\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{n+1}{y}^{2} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) a{b}^{2}{n}^{2}+6\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{2\,m+n+1} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) a{b}^{4}n+6\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-m}y \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) bn+8\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-m}y \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) b{m}^{3}+{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{2}^{{\frac{m}{2\,m+n+1}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-m}y \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) b{n}^{3}+4\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-m}y \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) b{n}^{3}+16\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-m}y \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) b{m}^{2}+16\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-m}y \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) bm+12\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-m}y \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) bn+6\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-m}y \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) b{n}^{2}+20\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-m}y \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) b{m}^{2}+12\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-m}y \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) b{n}^{2}+6\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{m+n+1}y \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) a{b}^{3}+3\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{n+1}{y}^{2} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) a{b}^{2}+10\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-2\,m}{y}^{2} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) mn+16\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-2\,m}{y}^{2} \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) mn+8\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-2\,m}{y}^{2} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ){m}^{2}n+10\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-m}y \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) bm+8\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-m}y \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) b{m}^{3}+5\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-2\,m}{y}^{2} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) m{n}^{2}+{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-2\,m}{y}^{2} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) +{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{2}^{{\frac{m}{2\,m+n+1}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-2\,m}{y}^{2} \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) +12\,{{\rm e}^{-6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{9}^{-{\frac{m}{2\,m+n+1}}}{3}^{-{\frac{n+1}{2\,m+n+1}}}{b}^{2}{m}^{2}n+16\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-m}y \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) bm{n}^{2}+16\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-m}y \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) b{m}^{2}n+32\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-m}y \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) bmn+20\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-m}y \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) bmn+10\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-m}y \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) bm{n}^{2}+20\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{-m}y \WhittakerM \left ({\frac{m+n+1}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) b{m}^{2}n+6\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{m+n+1}y \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) a{b}^{3}{n}^{2}+24\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{m+n+1}y \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) a{b}^{3}m+12\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{m+n+1}y \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) a{b}^{3}n+12\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{n+1}{y}^{2} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) a{b}^{2}m+12\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{n+1}{y}^{2} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) a{b}^{2}{m}^{2}+6\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{n+1}{y}^{2} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) a{b}^{2}n+24\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{m+n+1}y \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) a{b}^{3}{m}^{2}+12\,{{\rm e}^{-3\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}}}}{27}^{-{\frac{m}{2\,m+n+1}}}{9}^{-{\frac{n+1}{2\,m+n+1}}} \left ({\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) ^{-{\frac{m+n+1}{2\,m+n+1}}}{2}^{-{\frac{m+n+1}{2\,m+n+1}}}{x}^{2\,m+n+1} \WhittakerM \left ( -{\frac{m}{2\,m+n+1}},1/2\,{\frac{4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac{{b}^{2}{x}^{2\,m+n+1}a}{2\,m+n+1}} \right ) a{b}^{4}mn \right ) } \right ) \]

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39.16 problem number 16

problem number 285

Added January 2, 2019.

Problem 2.2.5.16 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a x^n y^3 + 3 a b x^{n+m} y^2+ c x^k y- 2 a b^3 x^{n+3 m} + b c x^{m+l} - b m x^{m-1} \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (-2 a b^3 x^{3 m+n}+3 a b y^2 x^{m+n}+a y^3 x^n+b c x^{l+m}-b m x^{m-1}+c y x^k\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ \text{ sol=() } \]

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39.17 problem number 17

problem number 286

Added January 2, 2019.

Problem 2.2.5.17 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a y^n + b x ^{\frac{n}{1-n}} \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (a y^n+b x^{\frac{n}{1-n}}\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int _{{\it \_b}}^{y}\!{1{x}^{{\frac{n}{n-1}}} \left ({{\it \_a}}^{n}{x}^{{\frac{n}{n-1}}}anx-a{{\it \_a}}^{n}{x}^{{\frac{n}{n-1}}}x+bxn+{x}^{{\frac{n}{n-1}}}{\it \_a}-bx \right ) ^{-1}}\,{\rm d}{\it \_a}n+\ln \left ( x \right ) +\int _{{\it \_b}}^{y}\!{1{x}^{{\frac{n}{n-1}}} \left ({{\it \_a}}^{n}{x}^{{\frac{n}{n-1}}}anx-a{{\it \_a}}^{n}{x}^{{\frac{n}{n-1}}}x+bxn+{x}^{{\frac{n}{n-1}}}{\it \_a}-bx \right ) ^{-1}}\,{\rm d}{\it \_a} \right ) \]

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39.18 problem number 18

problem number 287

Added January 2, 2019.

Problem 2.2.5.18 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a x^{m-n-(m n)} y^n + b x^m \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (a y^n x^{m (-n)+m-n}+b x^m\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( \int _{{\it \_b}}^{y}\!-{\frac{{x}^{mn}{x}^{n}}{{x}^{m}bx{x}^{mn}{x}^{n}-{x}^{mn}{x}^{n}{\it \_a}\,m+{x}^{m}xa{{\it \_a}}^{n}-{x}^{mn}{x}^{n}{\it \_a}}}\,{\rm d}{\it \_a}+\ln \left ( x \right ) \right ) \]

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39.19 problem number 19

problem number 288

Added January 2, 2019.

Problem 2.2.5.19 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a x^n y^k + b x^m y \right ) w_y = 0 \]

Mathematica

\[ \text{Timed out} \] Timed out

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{1}{b \left ( 2\,n{m}^{2}+3\,m{n}^{2}+{n}^{3}+2\,{m}^{2}+10\,mn+6\,{n}^{2}+7\,m+11\,n+6 \right ) } \left ( -2\,{{\rm e}^{{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}}{y}^{1-k}b{m}^{2}-6\,{{\rm e}^{{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}}{y}^{1-k}b{n}^{2}-7\,{{\rm e}^{{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}}{y}^{1-k}bm-11\,{{\rm e}^{{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}}{y}^{1-k}bn-{{\rm e}^{{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}}{y}^{1-k}b{n}^{3}+2\,{x}^{n-m} \left ( -{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac{m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac{-n+m}{m+1}},1/2\,{\frac{2\,m+3+n}{m+1}},-{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) a+4\,{x}^{n-m} \left ( -{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac{m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( 1/2\,{\frac{m+n+2}{m+1}},1/2\,{\frac{2\,m+3+n}{m+1}},-{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) a-2\,{{\rm e}^{{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}}{y}^{1-k}b{m}^{2}n-3\,{{\rm e}^{{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}}{y}^{1-k}bm{n}^{2}-10\,{{\rm e}^{{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}}{y}^{1-k}bmn- \left ( -{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac{m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac{-n+m}{m+1}},1/2\,{\frac{2\,m+3+n}{m+1}},-{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) abk{m}^{2}{x}^{n+1}-6\,{{\rm e}^{{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}}{y}^{1-k}b- \left ( -{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac{m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac{-n+m}{m+1}},1/2\,{\frac{2\,m+3+n}{m+1}},-{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) abk{x}^{n+1}+2\, \left ( -{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac{m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac{-n+m}{m+1}},1/2\,{\frac{2\,m+3+n}{m+1}},-{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) abm{x}^{n+1}+ \left ( -{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac{m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac{-n+m}{m+1}},1/2\,{\frac{2\,m+3+n}{m+1}},-{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ab{m}^{2}{x}^{n+1}+2\,{x}^{n-m} \left ( -{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac{m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac{-n+m}{m+1}},1/2\,{\frac{2\,m+3+n}{m+1}},-{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) amn+{x}^{n-m} \left ( -{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac{m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac{-n+m}{m+1}},1/2\,{\frac{2\,m+3+n}{m+1}},-{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) a{m}^{2}n+6\,{x}^{n-m} \left ( -{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac{m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( 1/2\,{\frac{m+n+2}{m+1}},1/2\,{\frac{2\,m+3+n}{m+1}},-{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) amn+{x}^{n-m} \left ( -{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac{m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( 1/2\,{\frac{m+n+2}{m+1}},1/2\,{\frac{2\,m+3+n}{m+1}},-{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) am{n}^{2}+2\,{x}^{n-m} \left ( -{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac{m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( 1/2\,{\frac{m+n+2}{m+1}},1/2\,{\frac{2\,m+3+n}{m+1}},-{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) a{m}^{2}n-2\, \left ( -{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac{m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac{-n+m}{m+1}},1/2\,{\frac{2\,m+3+n}{m+1}},-{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) abkm{x}^{n+1}+8\,{x}^{n-m} \left ( -{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac{m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( 1/2\,{\frac{m+n+2}{m+1}},1/2\,{\frac{2\,m+3+n}{m+1}},-{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) am+5\,{x}^{n-m} \left ( -{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac{m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( 1/2\,{\frac{m+n+2}{m+1}},1/2\,{\frac{2\,m+3+n}{m+1}},-{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) a{m}^{2}+{x}^{n-m} \left ( -{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac{m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( 1/2\,{\frac{m+n+2}{m+1}},1/2\,{\frac{2\,m+3+n}{m+1}},-{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) a{m}^{3}+4\,{x}^{n-m} \left ( -{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac{m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( 1/2\,{\frac{m+n+2}{m+1}},1/2\,{\frac{2\,m+3+n}{m+1}},-{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) an+{x}^{n-m} \left ( -{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac{m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( 1/2\,{\frac{m+n+2}{m+1}},1/2\,{\frac{2\,m+3+n}{m+1}},-{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) a{n}^{2}+ \left ( -{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac{m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac{-n+m}{m+1}},1/2\,{\frac{2\,m+3+n}{m+1}},-{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ab{x}^{n+1}+5\,{x}^{n-m} \left ( -{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac{m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac{-n+m}{m+1}},1/2\,{\frac{2\,m+3+n}{m+1}},-{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) am+4\,{x}^{n-m} \left ( -{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac{m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac{-n+m}{m+1}},1/2\,{\frac{2\,m+3+n}{m+1}},-{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) a{m}^{2}+{x}^{n-m} \left ( -{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac{m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac{-n+m}{m+1}},1/2\,{\frac{2\,m+3+n}{m+1}},-{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) a{m}^{3}+{x}^{n-m} \left ( -{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac{m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac{-n+m}{m+1}},1/2\,{\frac{2\,m+3+n}{m+1}},-{\frac{b{x}^{m+1} \left ( k-1 \right ) }{m+1}} \right ) an \right ) } \right ) \]

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39.20 problem number 20

problem number 289

Added January 2, 2019.

Problem 2.2.5.20 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a y^2 + b y+ c x^{2 b} \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{-\sqrt{a} y \sin \left (\frac{\sqrt{a} \sqrt{c} x^b}{b}\right )-\sqrt{c} x^b \cos \left (\frac{\sqrt{a} \sqrt{c} x^b}{b}\right )}{\sqrt{a} y \cos \left (\frac{\sqrt{a} \sqrt{c} x^b}{b}\right )-\sqrt{c} x^b \sin \left (\frac{\sqrt{a} \sqrt{c} x^b}{b}\right )}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{b} \left ({x}^{b}\sqrt{a}\sqrt{c}-\arctan \left ({\frac{{x}^{-b}\sqrt{a}y}{\sqrt{c}}} \right ) b \right ) } \right ) \]

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39.21 problem number 21

problem number 290

Added January 2, 2019.

Problem 2.2.5.21 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a y^2+(n+b x^n) y + c x^{2 n} \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{\exp \left (\frac{\sqrt{a} \sqrt{c} x^n \left (\frac{\sqrt{b^2-4 a c}}{\sqrt{a} \sqrt{c}}+\frac{b}{\sqrt{a} \sqrt{c}}\right )}{2 n}-\frac{\sqrt{a} \sqrt{c} x^n \left (\frac{b}{\sqrt{a} \sqrt{c}}-\frac{\sqrt{b^2-4 a c}}{\sqrt{a} \sqrt{c}}\right )}{2 n}\right ) \left (x^n \sqrt{b^2-4 a c}+2 a y+b x^n\right )}{x^n \sqrt{b^2-4 a c}-2 a y-b x^n}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{b}{\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }n} \left ( 2\,bn\arctan \left ({\frac{b \left ( 2\,{x}^{-n}ya+b \right ) }{\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }}} \right ) -\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{x}^{n} \right ) } \right ) \]

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39.22 problem number 22

problem number 291

Added January 2, 2019.

Problem 2.2.5.22 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a x^n y^2 + b y+ c x^{-n} \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{-2 a y x^{\frac{1}{2} \sqrt{a} \sqrt{c} \left (\frac{\sqrt{-4 a c+b^2+2 b n+n^2}}{\sqrt{a} \sqrt{c}}-\frac{-b-n}{\sqrt{a} \sqrt{c}}\right )+n}-b x^{\frac{1}{2} \sqrt{a} \sqrt{c} \left (\frac{\sqrt{-4 a c+b^2+2 b n+n^2}}{\sqrt{a} \sqrt{c}}-\frac{-b-n}{\sqrt{a} \sqrt{c}}\right )}-n x^{\frac{1}{2} \sqrt{a} \sqrt{c} \left (\frac{\sqrt{-4 a c+b^2+2 b n+n^2}}{\sqrt{a} \sqrt{c}}-\frac{-b-n}{\sqrt{a} \sqrt{c}}\right )}-\sqrt{-4 a c+b^2+2 b n+n^2} x^{\frac{1}{2} \sqrt{a} \sqrt{c} \left (\frac{\sqrt{-4 a c+b^2+2 b n+n^2}}{\sqrt{a} \sqrt{c}}-\frac{-b-n}{\sqrt{a} \sqrt{c}}\right )}}{2 a y x^{\frac{1}{2} \sqrt{a} \sqrt{c} \left (-\frac{\sqrt{-4 a c+b^2+2 b n+n^2}}{\sqrt{a} \sqrt{c}}-\frac{-b-n}{\sqrt{a} \sqrt{c}}\right )+n}+b x^{\frac{1}{2} \sqrt{a} \sqrt{c} \left (-\frac{\sqrt{-4 a c+b^2+2 b n+n^2}}{\sqrt{a} \sqrt{c}}-\frac{-b-n}{\sqrt{a} \sqrt{c}}\right )}+n x^{\frac{1}{2} \sqrt{a} \sqrt{c} \left (-\frac{\sqrt{-4 a c+b^2+2 b n+n^2}}{\sqrt{a} \sqrt{c}}-\frac{-b-n}{\sqrt{a} \sqrt{c}}\right )}-\sqrt{-4 a c+b^2+2 b n+n^2} x^{\frac{1}{2} \sqrt{a} \sqrt{c} \left (-\frac{\sqrt{-4 a c+b^2+2 b n+n^2}}{\sqrt{a} \sqrt{c}}-\frac{-b-n}{\sqrt{a} \sqrt{c}}\right )}}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{\sqrt{4\,ca-{b}^{2}-2\,bn-{n}^{2}}} \left ( \ln \left ( x \right ) \sqrt{4\,ca-{b}^{2}-2\,bn-{n}^{2}}-2\,\arctan \left ({\frac{2\,a{x}^{n}y+b+n}{\sqrt{4\,ca-{b}^{2}-2\,bn-{n}^{2}}}} \right ) \right ) } \right ) \]

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39.23 problem number 23

problem number 292

Added January 2, 2019.

Problem 2.2.5.23 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a x^n y^2+ m y- a b^2 x^{x+2 m} \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (-a b^2 x^{2 m+x}+a y^2 x^n+m y\right )+x w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{1 \left ( 2\,y\BesselI \left ( -{\frac{n+m}{n+x+2\,m}},2\,{\frac{ab{x}^{n/2}{x}^{x/2}{x}^{m}}{n+x+2\,m}} \right ){x}^{n}a+2\,x{\frac{\partial }{\partial x}}\BesselI \left ( -{\frac{n+m}{n+x+2\,m}},2\,{\frac{ab{x}^{n/2}{x}^{x/2}{x}^{m}}{n+x+2\,m}} \right ) +\BesselI \left ( -{\frac{n+m}{n+x+2\,m}},2\,{\frac{ab{x}^{n/2}{x}^{x/2}{x}^{m}}{n+x+2\,m}} \right ) m+\BesselI \left ( -{\frac{n+m}{n+x+2\,m}},2\,{\frac{ab{x}^{n/2}{x}^{x/2}{x}^{m}}{n+x+2\,m}} \right ) n \right ) \left ( 2\,y\BesselK \left ({\frac{n+m}{n+x+2\,m}},2\,{\frac{ab{x}^{n/2}{x}^{x/2}{x}^{m}}{n+x+2\,m}} \right ){x}^{n}a+2\,x{\frac{\partial }{\partial x}}\BesselK \left ({\frac{n+m}{n+x+2\,m}},2\,{\frac{ab{x}^{n/2}{x}^{x/2}{x}^{m}}{n+x+2\,m}} \right ) +\BesselK \left ({\frac{n+m}{n+x+2\,m}},2\,{\frac{ab{x}^{n/2}{x}^{x/2}{x}^{m}}{n+x+2\,m}} \right ) m+\BesselK \left ({\frac{n+m}{n+x+2\,m}},2\,{\frac{ab{x}^{n/2}{x}^{x/2}{x}^{m}}{n+x+2\,m}} \right ) n \right ) ^{-1}} \right ) \]

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39.24 problem number 24

problem number 293

Added January 2, 2019.

Problem 2.2.5.24 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( x^{2 n} y^2+(m-n) y+ x^{2 m} \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{m \tan ^{-1}\left (y x^{n-m}\right )+n \tan ^{-1}\left (y x^{n-m}\right )-x^{m+n}}{m+n}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{-\arctan \left ({x}^{n-m}y \right ) m-\arctan \left ({x}^{n-m}y \right ) n+{x}^{n+m}}{n+m}} \right ) \]

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39.25 problem number 25

problem number 294

Added January 2, 2019.

Problem 2.2.5.25 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a x^{2 n} y^2+ (b x^n -n) y + c \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{\exp \left (\frac{\sqrt{a} \sqrt{c} x^n \left (\frac{\sqrt{b^2-4 a c}}{\sqrt{a} \sqrt{c}}+\frac{b}{\sqrt{a} \sqrt{c}}\right )}{2 n}-\frac{\sqrt{a} \sqrt{c} x^n \left (\frac{b}{\sqrt{a} \sqrt{c}}-\frac{\sqrt{b^2-4 a c}}{\sqrt{a} \sqrt{c}}\right )}{2 n}\right ) \left (\sqrt{b^2-4 a c}+2 a y x^n+b\right )}{\sqrt{b^2-4 a c}-2 a y x^n-b}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{b}{\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }n} \left ( 2\,bn\arctan \left ({\frac{b \left ( 2\,a{x}^{n}y+b \right ) }{\sqrt{4\,ac{b}^{2}-{b}^{4}}}} \right ) -\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{x}^{n} \right ) } \right ) \]

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39.26 problem number 26

problem number 295

Added January 2, 2019.

Problem 2.2.5.26 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a x^{2 n + m} y^2 +(b x^{n+m}-n) y+ c x^m \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{\left (\sqrt{b^2-4 a c}+2 a y x^n+b\right ) \exp \left (\frac{\sqrt{a} \sqrt{c} \left (\frac{\sqrt{b^2-4 a c}}{\sqrt{a} \sqrt{c}}+\frac{b}{\sqrt{a} \sqrt{c}}\right ) x^{m+n}}{2 (m+n)}-\frac{\sqrt{a} \sqrt{c} \left (\frac{b}{\sqrt{a} \sqrt{c}}-\frac{\sqrt{b^2-4 a c}}{\sqrt{a} \sqrt{c}}\right ) x^{m+n}}{2 (m+n)}\right )}{\sqrt{b^2-4 a c}-2 a y x^n-b}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{b}{\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) } \left ( n+m \right ) } \left ( 2\,bm\arctan \left ({\frac{b \left ( 2\,a{x}^{n}y+b \right ) }{\sqrt{4\,ac{b}^{2}-{b}^{4}}}} \right ) +2\,bn\arctan \left ({\frac{b \left ( 2\,a{x}^{n}y+b \right ) }{\sqrt{4\,ac{b}^{2}-{b}^{4}}}} \right ) -\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{x}^{n}{x}^{m} \right ) } \right ) \]

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39.27 problem number 27

problem number 296

Added January 2, 2019.

Problem 2.2.5.27 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a y^3+3 a b x^n y^2 - b n x^n -2 a b^3 x^{3 n} \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{e^{-\frac{3 a b^2 x^{2 n}}{n}} \left (a y^2 e^{\frac{3 a b^2 x^{2 n}}{n}} \text{ExpIntegralEi}\left (-\frac{3 a b^2 x^{2 n}}{n}\right )+a b^2 x^{2 n} e^{\frac{3 a b^2 x^{2 n}}{n}} \text{ExpIntegralEi}\left (-\frac{3 a b^2 x^{2 n}}{n}\right )+2 a b y x^n e^{\frac{3 a b^2 x^{2 n}}{n}} \text{ExpIntegralEi}\left (-\frac{3 a b^2 x^{2 n}}{n}\right )+n\right )}{n \left (b x^n+y\right )^2}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{1}{n \left ({x}^{2\,n}{b}^{2}+2\,{x}^{n}by+{y}^{2} \right ) } \left ( \Ei \left ( 1,3\,{\frac{a{b}^{2}{x}^{2\,n}}{n}} \right ){x}^{2\,n}a{b}^{2}+2\,y\Ei \left ( 1,3\,{\frac{a{b}^{2}{x}^{2\,n}}{n}} \right ){x}^{n}ab+{y}^{2}\Ei \left ( 1,3\,{\frac{a{b}^{2}{x}^{2\,n}}{n}} \right ) a-{{\rm e}^{-3\,{\frac{a{b}^{2}{x}^{2\,n}}{n}}}}n \right ) } \right ) \]

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39.28 problem number 28

problem number 297

Added January 2, 2019.

Problem 2.2.5.28 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a x^{2 n +1} y^3 + (b x -n) y + c x^{1-n} \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (a y^3 x^{2 n+1}+y (b x-n)+c x^{1-n}\right )+x w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({b}^{3}\sum _{{\it \_R}=\RootOf \left ({{\it \_Z}}^{3}a{c}^{2}+{\it \_Z}\,{b}^{3}-{b}^{3} \right ) }{\frac{1}{3\,{{\it \_R}}^{2}a{c}^{2}+{b}^{3}}\ln \left ( -{\frac{{x}^{n}by+{\it \_R}\,c}{c}} \right ) }-bx \right ) \] Solution contains RootOf

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39.29 problem number 29

problem number 298

Added January 2, 2019.

Problem 2.2.5.29 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a x^{n+2} y^3+ (b x^n-1) y + c x^{n-1} \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (a y^3 x^{n+2}+y \left (b x^n-1\right )+c x^{n-1}\right )+x w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{n} \left ({b}^{3}\sum _{{\it \_R}=\RootOf \left ({{\it \_Z}}^{3}a{c}^{2}+{\it \_Z}\,{b}^{3}-{b}^{3} \right ) }{\frac{1}{3\,{{\it \_R}}^{2}a{c}^{2}+{b}^{3}}\ln \left ( -{\frac{bxy+{\it \_R}\,c}{c}} \right ) }n-b{x}^{n} \right ) } \right ) \] Solution contains RootOf

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39.30 problem number 30

problem number 299

Added January 2, 2019.

Problem 2.2.5.30 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( y+a x^{n - m }y^m+b x^{n-k} y^k \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (a y^m x^{n-m}+b y^k x^{n-k}+y\right )+x w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{ \left ( n-1 \right ) x} \left ( \int _{{\it \_b}}^{y}\!-{\frac{{x}^{m}{x}^{k}}{x \left ( b{x}^{m}{{\it \_a}}^{k}+{{\it \_a}}^{m}{x}^{k}a \right ) }}\,{\rm d}{\it \_a}xn-\int _{{\it \_b}}^{y}\!-{\frac{{x}^{m}{x}^{k}}{x \left ( b{x}^{m}{{\it \_a}}^{k}+{{\it \_a}}^{m}{x}^{k}a \right ) }}\,{\rm d}{\it \_a}x+{x}^{n} \right ) } \right ) \]

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39.31 problem number 31

problem number 300

Added January 2, 2019.

Problem 2.2.5.31 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ y w_x + \left (x^{n-1}((1+2 n)x+a n) y-n x^{2 n}(x+a) \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (y x^{n-1} (a n+(2 n+1) x)-n (a+x) x^{2 n}\right )+y w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( 1/2\,{\frac{1}{x} \left ( 2\,{{\rm e}^{2\,{\frac{1}{\sqrt{-{n}^{2}}}\arctan \left ({\frac{ \left ( -2\,{x}^{n}a-{x}^{n+1}+y \right ) n}{\sqrt{-{n}^{2}} \left ( -{x}^{n+1}+y \right ) }} \right ) }}}na+{{\rm e}^{2\,{\frac{1}{\sqrt{-{n}^{2}}}\arctan \left ({\frac{ \left ( -2\,{x}^{n}a-{x}^{n+1}+y \right ) n}{\sqrt{-{n}^{2}} \left ( -{x}^{n+1}+y \right ) }} \right ) }}}nx-\int ^{-2\,{\frac{1}{\sqrt{-{n}^{2}}}\arctan \left ({\frac{ \left ( -2\,{x}^{n}a-{x}^{n+1}+y \right ) n}{\sqrt{-{n}^{2}} \left ( -{x}^{n+1}+y \right ) }} \right ) }}\!\tan \left ( 1/2\,{\it \_a}\,\sqrt{-{n}^{2}} \right ){{\rm e}^{-{\it \_a}}}{d{\it \_a}}\sqrt{-{n}^{2}}x \right ) } \right ) \]

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39.32 problem number 32

problem number 301

Added January 2, 2019.

Problem 2.2.5.32 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ y w_x + \left ( (a(2 n +k)x^k+b)x^{n-1}y -(a^2 n x^{2 k}+ a b x^k -c) x^{2 n-1} \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (y x^{n-1} \left (a (k+2 n) x^k+b\right )-x^{2 n-1} \left (a^2 n x^{2 k}+a b x^k-c\right )\right )+y w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ \text{ sol=() } \]

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39.33 problem number 33

problem number 302

Added January 2, 2019.

Problem 2.2.5.33 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x(2 a x y+b) w_x - \left ( a(m+3) x y^2+b(m+2)y-c x^m \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (-\frac{x^{m+2} \left (-2 a m x y^2-2 a x y^2-2 b m y-2 b y+c x^m\right )}{2 a (m+1)}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac{ \left ( 2\,ax{y}^{2}m+2\,ax{y}^{2}+2\,bym+2\,by-c{x}^{m} \right ){x}^{2}{x}^{m}}{m+1}} \right ) \]

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39.34 problem number 34

problem number 303

Added January 2, 2019.

Problem 2.2.5.34 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x^2(2 a x y+b) w_x - \left ( 4 a x^2 y^2 + 3 b x y-c x^2 - k \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{4 a x^4 y^2+4 b x^3 y-c x^4-2 k x^2}{4 a}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -a{x}^{4}{y}^{2}-b{x}^{3}y+1/4\,c{x}^{4}+1/2\,{x}^{2}k \right ) \]

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39.35 problem number 35

problem number 304

Added January 2, 2019.

Problem 2.2.5.35 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a x^m w_x + b y^n w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (-\frac{x^{-m} y^{-n} \left (-a y x^m+a m y x^m+b x y^n-b n x y^n\right )}{a (m-1) (n-1)}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{{y}^{-n+1}am-{y}^{-n+1}a-{x}^{-m+1}bn+b{x}^{-m+1}}{a \left ( m-1 \right ) }} \right ) \]

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39.36 problem number 36

problem number 305

Added January 2, 2019.

Problem 2.2.5.36 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a x^n w_x + (b y+ c x^m) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{b^{-\frac{n}{n-1}} (a-a n)^{-\frac{m}{n-1}-\frac{1}{n-1}} e^{-\frac{b x^{1-n}}{a (1-n)}} \left (-c (a-a n)^{\frac{n}{n-1}} b^{\frac{m}{n-1}+\frac{1}{n-1}} e^{\frac{b x^{1-n}}{a (1-n)}} \text{Gamma}\left (\frac{-m+n-1}{n-1},\frac{b x^{1-n}}{a-a n}\right )-a y b^{\frac{n}{n-1}} (a-a n)^{\frac{m}{n-1}+\frac{1}{n-1}}+a n y b^{\frac{n}{n-1}} (a-a n)^{\frac{m}{n-1}+\frac{1}{n-1}}\right )}{a (n-1)}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{ab \left ({m}^{3}-6\,n{m}^{2}+11\,m{n}^{2}-6\,{n}^{3}+6\,{m}^{2}-22\,mn+18\,{n}^{2}+11\,m-18\,n+6 \right ) } \left ( -{{\rm e}^{1/2\,{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}{x}^{-n+m+1} \left ( -{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac{m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac{m}{n-1}},-1/2\,{\frac{m-3\,n+3}{n-1}},-{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) bc-2\,{{\rm e}^{1/2\,{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac{m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac{m}{n-1}},-1/2\,{\frac{m-3\,n+3}{n-1}},-{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ac-4\,{{\rm e}^{1/2\,{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac{m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac{m-2\,n+2}{n-1}},-1/2\,{\frac{m-3\,n+3}{n-1}},-{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ac+11\,{{\rm e}^{{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}yabm+6\,{{\rm e}^{{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}yab{m}^{2}+18\,{{\rm e}^{{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}yab{n}^{2}-18\,{{\rm e}^{{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}yabn+{{\rm e}^{{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}yab{m}^{3}-6\,{{\rm e}^{{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}yab{n}^{3}-6\,{{\rm e}^{{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}yab{m}^{2}n+11\,{{\rm e}^{{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}yabm{n}^{2}-22\,{{\rm e}^{{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}yabmn+6\,ya{{\rm e}^{{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}b+6\,{{\rm e}^{1/2\,{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac{m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac{m}{n-1}},-1/2\,{\frac{m-3\,n+3}{n-1}},-{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) acn-{{\rm e}^{1/2\,{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}{x}^{-n+m+1} \left ( -{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac{m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac{m}{n-1}},-1/2\,{\frac{m-3\,n+3}{n-1}},-{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) bc{n}^{2}+2\,{{\rm e}^{1/2\,{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}{x}^{-n+m+1} \left ( -{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac{m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac{m}{n-1}},-1/2\,{\frac{m-3\,n+3}{n-1}},-{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) bcn+12\,{{\rm e}^{1/2\,{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac{m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac{m-2\,n+2}{n-1}},-1/2\,{\frac{m-3\,n+3}{n-1}},-{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) acn+2\,{{\rm e}^{1/2\,{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac{m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac{m}{n-1}},-1/2\,{\frac{m-3\,n+3}{n-1}},-{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ac{n}^{3}+4\,{{\rm e}^{1/2\,{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac{m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac{m-2\,n+2}{n-1}},-1/2\,{\frac{m-3\,n+3}{n-1}},-{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ac{n}^{3}-6\,{{\rm e}^{1/2\,{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac{m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac{m}{n-1}},-1/2\,{\frac{m-3\,n+3}{n-1}},-{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ac{n}^{2}-{{\rm e}^{1/2\,{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac{m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac{m-2\,n+2}{n-1}},-1/2\,{\frac{m-3\,n+3}{n-1}},-{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ac{m}^{2}-12\,{{\rm e}^{1/2\,{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac{m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac{m-2\,n+2}{n-1}},-1/2\,{\frac{m-3\,n+3}{n-1}},-{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ac{n}^{2}-{{\rm e}^{1/2\,{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac{m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac{m}{n-1}},-1/2\,{\frac{m-3\,n+3}{n-1}},-{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) acm-4\,{{\rm e}^{1/2\,{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac{m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac{m-2\,n+2}{n-1}},-1/2\,{\frac{m-3\,n+3}{n-1}},-{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) acm-{{\rm e}^{1/2\,{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac{m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac{m}{n-1}},-1/2\,{\frac{m-3\,n+3}{n-1}},-{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) acm{n}^{2}+{{\rm e}^{1/2\,{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac{m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac{m-2\,n+2}{n-1}},-1/2\,{\frac{m-3\,n+3}{n-1}},-{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ac{m}^{2}n-4\,{{\rm e}^{1/2\,{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac{m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac{m-2\,n+2}{n-1}},-1/2\,{\frac{m-3\,n+3}{n-1}},-{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) acm{n}^{2}+2\,{{\rm e}^{1/2\,{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac{m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac{m}{n-1}},-1/2\,{\frac{m-3\,n+3}{n-1}},-{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) acmn+8\,{{\rm e}^{1/2\,{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac{m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac{m-2\,n+2}{n-1}},-1/2\,{\frac{m-3\,n+3}{n-1}},-{\frac{b{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) acmn \right ) } \right ) \]

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39.37 problem number 37

problem number 306

Added January 2, 2019.

Problem 2.2.5.37 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a x^k w_x + (y^n+ b x^m y) w_y = 0 \]

Mathematica

\[ \text{Timed out} \] Timed out

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{a} \left ({ \left ({y}^{-1} \right ) ^{ \left ( k-m-1 \right ) ^{-1}}a \left ({y}^{-1} \right ) ^{{\frac{kn}{k-m-1}}} \left ({y}^{-1} \right ) ^{{\frac{m}{k-m-1}}}{{\rm e}^{{\frac{b{x}^{-k+m+1}}{ \left ( k-m-1 \right ) a}}}} \left ( \left ({y}^{-1} \right ) ^{{\frac{mn}{k-m-1}}} \right ) ^{-1} \left ( \left ({y}^{-1} \right ) ^{{\frac{n}{k-m-1}}} \right ) ^{-1} \left ({{\rm e}^{{\frac{b{x}^{-k+m+1}n}{ \left ( k-m-1 \right ) a}}}} \right ) ^{-1} \left ( \left ({y}^{-1} \right ) ^{{\frac{k}{k-m-1}}} \right ) ^{-1}}+{\frac{n}{-k+m+1} \left ({\frac{b \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{{\frac{k}{-k+m+1}}- \left ( -k+m+1 \right ) ^{-1}} \left ({\frac{ \left ( k-m-1 \right ) ^{2}a}{ \left ( k-1 \right ) \left ( 2\,k-2-m \right ) \left ( 3\,k-3-2\,m \right ) b \left ( n-1 \right ) }{x}^{{\frac{{k}^{2}}{-k+m+1}}-{\frac{mk}{-k+m+1}}-2\,{\frac{k}{-k+m+1}}+{\frac{m}{-k+m+1}}+ \left ( -k+m+1 \right ) ^{-1}+k-m-1} \left ({\frac{b \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{-{\frac{k}{-k+m+1}}+ \left ( -k+m+1 \right ) ^{-1}} \left ({\frac{b{x}^{-k+m+1} \left ( n-1 \right ){k}^{2}}{ \left ( k-m-1 \right ) a}}-2\,{\frac{b{x}^{-k+m+1} \left ( n-1 \right ) km}{ \left ( k-m-1 \right ) a}}+{\frac{b{x}^{-k+m+1} \left ( n-1 \right ){m}^{2}}{ \left ( k-m-1 \right ) a}}-2\,{\frac{b{x}^{-k+m+1} \left ( n-1 \right ) k}{ \left ( k-m-1 \right ) a}}+2\,{\frac{b{x}^{-k+m+1} \left ( n-1 \right ) m}{ \left ( k-m-1 \right ) a}}+2\,{k}^{2}-3\,mk+{m}^{2}+{\frac{b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}}-4\,k+3\,m+2 \right ) \left ({\frac{b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{-1/2\,{\frac{2\,k-2-m}{k-m-1}}}{{\rm e}^{-1/2\,{\frac{b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}}}} \WhittakerM \left ({\frac{k-1}{k-m-1}}-1/2\,{\frac{2\,k-2-m}{k-m-1}},1/2\,{\frac{2\,k-2-m}{k-m-1}}+1/2,{\frac{b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) }+{\frac{ \left ( k-m-1 \right ) ^{2} \left ( 2\,k-2-m \right ) a}{ \left ( k-1 \right ) \left ( 3\,k-3-2\,m \right ) b \left ( n-1 \right ) }{x}^{{\frac{{k}^{2}}{-k+m+1}}-{\frac{mk}{-k+m+1}}-2\,{\frac{k}{-k+m+1}}+{\frac{m}{-k+m+1}}+ \left ( -k+m+1 \right ) ^{-1}+k-m-1} \left ({\frac{b \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{-{\frac{k}{-k+m+1}}+ \left ( -k+m+1 \right ) ^{-1}} \left ({\frac{b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{-1/2\,{\frac{2\,k-2-m}{k-m-1}}}{{\rm e}^{-1/2\,{\frac{b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}}}} \WhittakerM \left ({\frac{k-1}{k-m-1}}-1/2\,{\frac{2\,k-2-m}{k-m-1}}+1,1/2\,{\frac{2\,k-2-m}{k-m-1}}+1/2,{\frac{b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) } \right ) }-{\frac{1}{-k+m+1} \left ({\frac{b \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{{\frac{k}{-k+m+1}}- \left ( -k+m+1 \right ) ^{-1}} \left ({\frac{ \left ( k-m-1 \right ) ^{2}a}{ \left ( k-1 \right ) \left ( 2\,k-2-m \right ) \left ( 3\,k-3-2\,m \right ) b \left ( n-1 \right ) }{x}^{{\frac{{k}^{2}}{-k+m+1}}-{\frac{mk}{-k+m+1}}-2\,{\frac{k}{-k+m+1}}+{\frac{m}{-k+m+1}}+ \left ( -k+m+1 \right ) ^{-1}+k-m-1} \left ({\frac{b \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{-{\frac{k}{-k+m+1}}+ \left ( -k+m+1 \right ) ^{-1}} \left ({\frac{b{x}^{-k+m+1} \left ( n-1 \right ){k}^{2}}{ \left ( k-m-1 \right ) a}}-2\,{\frac{b{x}^{-k+m+1} \left ( n-1 \right ) km}{ \left ( k-m-1 \right ) a}}+{\frac{b{x}^{-k+m+1} \left ( n-1 \right ){m}^{2}}{ \left ( k-m-1 \right ) a}}-2\,{\frac{b{x}^{-k+m+1} \left ( n-1 \right ) k}{ \left ( k-m-1 \right ) a}}+2\,{\frac{b{x}^{-k+m+1} \left ( n-1 \right ) m}{ \left ( k-m-1 \right ) a}}+2\,{k}^{2}-3\,mk+{m}^{2}+{\frac{b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}}-4\,k+3\,m+2 \right ) \left ({\frac{b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{-1/2\,{\frac{2\,k-2-m}{k-m-1}}}{{\rm e}^{-1/2\,{\frac{b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}}}} \WhittakerM \left ({\frac{k-1}{k-m-1}}-1/2\,{\frac{2\,k-2-m}{k-m-1}},1/2\,{\frac{2\,k-2-m}{k-m-1}}+1/2,{\frac{b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) }+{\frac{ \left ( k-m-1 \right ) ^{2} \left ( 2\,k-2-m \right ) a}{ \left ( k-1 \right ) \left ( 3\,k-3-2\,m \right ) b \left ( n-1 \right ) }{x}^{{\frac{{k}^{2}}{-k+m+1}}-{\frac{mk}{-k+m+1}}-2\,{\frac{k}{-k+m+1}}+{\frac{m}{-k+m+1}}+ \left ( -k+m+1 \right ) ^{-1}+k-m-1} \left ({\frac{b \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{-{\frac{k}{-k+m+1}}+ \left ( -k+m+1 \right ) ^{-1}} \left ({\frac{b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{-1/2\,{\frac{2\,k-2-m}{k-m-1}}}{{\rm e}^{-1/2\,{\frac{b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}}}} \WhittakerM \left ({\frac{k-1}{k-m-1}}-1/2\,{\frac{2\,k-2-m}{k-m-1}}+1,1/2\,{\frac{2\,k-2-m}{k-m-1}}+1/2,{\frac{b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) } \right ) } \right ) } \right ) \]

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39.38 problem number 38

problem number 307

Added January 2, 2019.

Problem 2.2.5.38 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x(a x^k+b) w_x + \left ( \alpha x^n y^2+(\beta -a n x^k)y+\gamma x^{-n} \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{-\beta \exp \left (\frac{\sqrt{\alpha } \sqrt{\gamma } \left (k \log (x)-\log \left (a x^k+b\right )\right ) \left (\sqrt{\frac{(-b n-\beta )^2}{\alpha \gamma }-4}-\frac{-b n-\beta }{\sqrt{\alpha } \sqrt{\gamma }}\right )}{2 b k}\right )-b n \exp \left (\frac{\sqrt{\alpha } \sqrt{\gamma } \left (k \log (x)-\log \left (a x^k+b\right )\right ) \left (\sqrt{\frac{(-b n-\beta )^2}{\alpha \gamma }-4}-\frac{-b n-\beta }{\sqrt{\alpha } \sqrt{\gamma }}\right )}{2 b k}\right )-\sqrt{\alpha } \sqrt{\gamma } \sqrt{\frac{(-b n-\beta )^2}{\alpha \gamma }-4} \exp \left (\frac{\sqrt{\alpha } \sqrt{\gamma } \left (k \log (x)-\log \left (a x^k+b\right )\right ) \left (\sqrt{\frac{(-b n-\beta )^2}{\alpha \gamma }-4}-\frac{-b n-\beta }{\sqrt{\alpha } \sqrt{\gamma }}\right )}{2 b k}\right )-2 \alpha y x^n \exp \left (\frac{\sqrt{\alpha } \sqrt{\gamma } \left (k \log (x)-\log \left (a x^k+b\right )\right ) \left (\sqrt{\frac{(-b n-\beta )^2}{\alpha \gamma }-4}-\frac{-b n-\beta }{\sqrt{\alpha } \sqrt{\gamma }}\right )}{2 b k}\right )}{\beta \exp \left (\frac{\sqrt{\alpha } \sqrt{\gamma } \left (k \log (x)-\log \left (a x^k+b\right )\right ) \left (-\frac{-b n-\beta }{\sqrt{\alpha } \sqrt{\gamma }}-\sqrt{\frac{(-b n-\beta )^2}{\alpha \gamma }-4}\right )}{2 b k}\right )+b n \exp \left (\frac{\sqrt{\alpha } \sqrt{\gamma } \left (k \log (x)-\log \left (a x^k+b\right )\right ) \left (-\frac{-b n-\beta }{\sqrt{\alpha } \sqrt{\gamma }}-\sqrt{\frac{(-b n-\beta )^2}{\alpha \gamma }-4}\right )}{2 b k}\right )-\sqrt{\alpha } \sqrt{\gamma } \sqrt{\frac{(-b n-\beta )^2}{\alpha \gamma }-4} \exp \left (\frac{\sqrt{\alpha } \sqrt{\gamma } \left (k \log (x)-\log \left (a x^k+b\right )\right ) \left (-\frac{-b n-\beta }{\sqrt{\alpha } \sqrt{\gamma }}-\sqrt{\frac{(-b n-\beta )^2}{\alpha \gamma }-4}\right )}{2 b k}\right )+2 \alpha y x^n \exp \left (\frac{\sqrt{\alpha } \sqrt{\gamma } \left (k \log (x)-\log \left (a x^k+b\right )\right ) \left (-\frac{-b n-\beta }{\sqrt{\alpha } \sqrt{\gamma }}-\sqrt{\frac{(-b n-\beta )^2}{\alpha \gamma }-4}\right )}{2 b k}\right )}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{\sqrt{- \left ( bn+\beta \right ) ^{2} \left ( -{b}^{2}{n}^{2}-2\,b\beta \,n+4\,\alpha \,g-{\beta }^{2} \right ) }bk} \left ( -2\,{b}^{3}k{n}^{2}\arctanh \left ({\frac{2\,y{x}^{n}\alpha \,bn+2\,y{x}^{n}\alpha \,\beta +{b}^{2}{n}^{2}+2\,b\beta \,n+{\beta }^{2}}{\sqrt{- \left ( bn+\beta \right ) ^{2} \left ( -{b}^{2}{n}^{2}-2\,b\beta \,n+4\,\alpha \,g-{\beta }^{2} \right ) }}} \right ) -4\,{b}^{2}\beta \,kn\arctanh \left ({\frac{2\,y{x}^{n}\alpha \,bn+2\,y{x}^{n}\alpha \,\beta +{b}^{2}{n}^{2}+2\,b\beta \,n+{\beta }^{2}}{\sqrt{- \left ( bn+\beta \right ) ^{2} \left ( -{b}^{2}{n}^{2}-2\,b\beta \,n+4\,\alpha \,g-{\beta }^{2} \right ) }}} \right ) -\sqrt{- \left ( bn+\beta \right ) ^{2} \left ( -{b}^{2}{n}^{2}-2\,b\beta \,n+4\,\alpha \,g-{\beta }^{2} \right ) }\ln \left ( x \right ) bkn-2\,b{\beta }^{2}k\arctanh \left ({\frac{2\,y{x}^{n}\alpha \,bn+2\,y{x}^{n}\alpha \,\beta +{b}^{2}{n}^{2}+2\,b\beta \,n+{\beta }^{2}}{\sqrt{- \left ( bn+\beta \right ) ^{2} \left ( -{b}^{2}{n}^{2}-2\,b\beta \,n+4\,\alpha \,g-{\beta }^{2} \right ) }}} \right ) +\ln \left ({x}^{k}a+b \right ) \sqrt{- \left ( bn+\beta \right ) ^{2} \left ( -{b}^{2}{n}^{2}-2\,b\beta \,n+4\,\alpha \,g-{\beta }^{2} \right ) }bn-\sqrt{- \left ( bn+\beta \right ) ^{2} \left ( -{b}^{2}{n}^{2}-2\,b\beta \,n+4\,\alpha \,g-{\beta }^{2} \right ) }\ln \left ( x \right ) \beta \,k+\ln \left ({x}^{k}a+b \right ) \sqrt{- \left ( bn+\beta \right ) ^{2} \left ( -{b}^{2}{n}^{2}-2\,b\beta \,n+4\,\alpha \,g-{\beta }^{2} \right ) }\beta \right ) } \right ) \]

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39.39 problem number 39

problem number 308

Added January 2, 2019.

Problem 2.2.5.39 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (y+ A x^n + a) w_x - \left ( n A x^{n-1} y + k x^m + b\right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{2 a m y+2 a y+2 A m y x^n+2 A y x^n+2 b m x+2 b x+2 k x^{m+1}+m y^2+y^2}{m+1}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac{2\,Ay{x}^{n}m+2\,A{x}^{n}y+{y}^{2}m+2\,aym+2\,k{x}^{m}x+2\,bxm+{y}^{2}+2\,ay+2\,bx}{m+1}} \right ) \]

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39.40 problem number 40

problem number 309

Added January 2, 2019.

Problem 2.2.5.40 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (y+ a x^{n+1}+b x^n) w_x + \left (a n x^n + c x^{n-1} \right ) y w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(1,0)}(x,y) \left (a x^{n+1}+b x^n+y\right )+y w^{(0,1)}(x,y) \left (a n x^n+c x^{n-1}\right )=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ \text{ sol=() } \]

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39.41 problem number 41

problem number 310

Added January 2, 2019.

Problem 2.2.5.41 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x(2 a x^n y+b) w_x - \left (a(3 n+m)x^n y^2+b(2 n+m)y-A x^m -C x^{-n} \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{x^{m+n} \left (2 a m y^2 x^{2 n}+2 a n y^2 x^{2 n}-A x^{m+n}+2 b m y x^n+2 b n y x^n-2 \text{C0}\right )}{2 a (m+n)}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( 1/2\,{\frac{-2\,{x}^{3\,n+m}{y}^{2}am-2\,{x}^{3\,n+m}{y}^{2}an-2\,{x}^{m+2\,n}ybm-2\,{x}^{m+2\,n}ybn+A{x}^{2\,n+2\,m}+2\,{x}^{n+m}C}{n+m}} \right ) \]

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39.42 problem number 42

problem number 311

Added January 2, 2019.

Problem 2.2.5.42 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a x^n+b x^2+ x y) w_x + \left (c x^n + b x y+ y^2 \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(1,0)}(x,y) \left (a x^n+b x^2+x y\right )+w^{(0,1)}(x,y) \left (b x y+c x^n+y^2\right )=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( 1/3\,{\frac{1}{n-2} \left ( \ln \left ( ab{x}^{2}+{a}^{2}{x}^{n}+c{x}^{2} \right ){n}^{2}-2\,\ln \left ( x \right ){n}^{2}-3\,\ln \left ( ab{x}^{2}+{a}^{2}{x}^{n}+c{x}^{2} \right ) n+{\frac{n}{ \left ( n-2 \right ) \left ( n-1 \right ) } \left ( \ln \left ( -9\,{\frac{ \left ( a{n}^{2}y-c{n}^{2}x-3\,any+3\,cnx+3\,ay-3\,cx \right ) x}{ \left ({x}^{n}a+b{x}^{2}+yx \right ) an}} \right ){n}^{3}-{n}^{3}\ln \left ( 9\,{\frac{ab{x}^{2}{n}^{2}+{n}^{2}{a}^{2}{x}^{n}-3\,abn{x}^{2}+c{x}^{2}{n}^{2}-3\,{a}^{2}{x}^{n}n+3\,ab{x}^{2}-3\,cn{x}^{2}+3\,{a}^{2}{x}^{n}+3\,c{x}^{2}}{ \left ( n{x}^{2}b+nxy+{x}^{n}an-3\,b{x}^{2}-3\,yx-3\,{x}^{n}a \right ) a}} \right ) +{n}^{2}\ln \left ( 9\,{\frac{{n}^{3}{x}^{2}ba+{n}^{3}xya+{x}^{n}{n}^{3}{a}^{2}-4\,ab{x}^{2}{n}^{2}-5\,a{n}^{2}xy-4\,{n}^{2}{a}^{2}{x}^{n}+6\,abn{x}^{2}+c{x}^{2}{n}^{2}+9\,anxy+6\,{a}^{2}{x}^{n}n-3\,ab{x}^{2}-3\,cn{x}^{2}-6\,axy-3\,{a}^{2}{x}^{n}+3\,c{x}^{2}}{ \left ( 2\,n{x}^{2}b+2\,nxy+2\,{x}^{n}an-3\,b{x}^{2}-3\,yx-3\,{x}^{n}a \right ) a}} \right ) -5\,\ln \left ( -9\,{\frac{ \left ( a{n}^{2}y-c{n}^{2}x-3\,any+3\,cnx+3\,ay-3\,cx \right ) x}{ \left ({x}^{n}a+b{x}^{2}+yx \right ) an}} \right ){n}^{2}+4\,{n}^{2}\ln \left ( 9\,{\frac{ab{x}^{2}{n}^{2}+{n}^{2}{a}^{2}{x}^{n}-3\,abn{x}^{2}+c{x}^{2}{n}^{2}-3\,{a}^{2}{x}^{n}n+3\,ab{x}^{2}-3\,cn{x}^{2}+3\,{a}^{2}{x}^{n}+3\,c{x}^{2}}{ \left ( n{x}^{2}b+nxy+{x}^{n}an-3\,b{x}^{2}-3\,yx-3\,{x}^{n}a \right ) a}} \right ) -3\,n\ln \left ( 9\,{\frac{{n}^{3}{x}^{2}ba+{n}^{3}xya+{x}^{n}{n}^{3}{a}^{2}-4\,ab{x}^{2}{n}^{2}-5\,a{n}^{2}xy-4\,{n}^{2}{a}^{2}{x}^{n}+6\,abn{x}^{2}+c{x}^{2}{n}^{2}+9\,anxy+6\,{a}^{2}{x}^{n}n-3\,ab{x}^{2}-3\,cn{x}^{2}-6\,axy-3\,{a}^{2}{x}^{n}+3\,c{x}^{2}}{ \left ( 2\,n{x}^{2}b+2\,nxy+2\,{x}^{n}an-3\,b{x}^{2}-3\,yx-3\,{x}^{n}a \right ) a}} \right ) +9\,\ln \left ( -9\,{\frac{ \left ( a{n}^{2}y-c{n}^{2}x-3\,any+3\,cnx+3\,ay-3\,cx \right ) x}{ \left ({x}^{n}a+b{x}^{2}+yx \right ) an}} \right ) n-6\,n\ln \left ( 9\,{\frac{ab{x}^{2}{n}^{2}+{n}^{2}{a}^{2}{x}^{n}-3\,abn{x}^{2}+c{x}^{2}{n}^{2}-3\,{a}^{2}{x}^{n}n+3\,ab{x}^{2}-3\,cn{x}^{2}+3\,{a}^{2}{x}^{n}+3\,c{x}^{2}}{ \left ( n{x}^{2}b+nxy+{x}^{n}an-3\,b{x}^{2}-3\,yx-3\,{x}^{n}a \right ) a}} \right ) +3\,\ln \left ( 9\,{\frac{{n}^{3}{x}^{2}ba+{n}^{3}xya+{x}^{n}{n}^{3}{a}^{2}-4\,ab{x}^{2}{n}^{2}-5\,a{n}^{2}xy-4\,{n}^{2}{a}^{2}{x}^{n}+6\,abn{x}^{2}+c{x}^{2}{n}^{2}+9\,anxy+6\,{a}^{2}{x}^{n}n-3\,ab{x}^{2}-3\,cn{x}^{2}-6\,axy-3\,{a}^{2}{x}^{n}+3\,c{x}^{2}}{ \left ( 2\,n{x}^{2}b+2\,nxy+2\,{x}^{n}an-3\,b{x}^{2}-3\,yx-3\,{x}^{n}a \right ) a}} \right ) -6\,\ln \left ( -9\,{\frac{ \left ( a{n}^{2}y-c{n}^{2}x-3\,any+3\,cnx+3\,ay-3\,cx \right ) x}{ \left ({x}^{n}a+b{x}^{2}+yx \right ) an}} \right ) +3\,\ln \left ( 9\,{\frac{ab{x}^{2}{n}^{2}+{n}^{2}{a}^{2}{x}^{n}-3\,abn{x}^{2}+c{x}^{2}{n}^{2}-3\,{a}^{2}{x}^{n}n+3\,ab{x}^{2}-3\,cn{x}^{2}+3\,{a}^{2}{x}^{n}+3\,c{x}^{2}}{ \left ( n{x}^{2}b+nxy+{x}^{n}an-3\,b{x}^{2}-3\,yx-3\,{x}^{n}a \right ) a}} \right ) \right ) }+6\,n\ln \left ( x \right ) +3\,\ln \left ( ab{x}^{2}+{a}^{2}{x}^{n}+c{x}^{2} \right ) -2\,{\frac{1}{ \left ( n-2 \right ) \left ( n-1 \right ) } \left ( \ln \left ( -9\,{\frac{ \left ( a{n}^{2}y-c{n}^{2}x-3\,any+3\,cnx+3\,ay-3\,cx \right ) x}{ \left ({x}^{n}a+b{x}^{2}+yx \right ) an}} \right ){n}^{3}-{n}^{3}\ln \left ( 9\,{\frac{ab{x}^{2}{n}^{2}+{n}^{2}{a}^{2}{x}^{n}-3\,abn{x}^{2}+c{x}^{2}{n}^{2}-3\,{a}^{2}{x}^{n}n+3\,ab{x}^{2}-3\,cn{x}^{2}+3\,{a}^{2}{x}^{n}+3\,c{x}^{2}}{ \left ( n{x}^{2}b+nxy+{x}^{n}an-3\,b{x}^{2}-3\,yx-3\,{x}^{n}a \right ) a}} \right ) +{n}^{2}\ln \left ( 9\,{\frac{{n}^{3}{x}^{2}ba+{n}^{3}xya+{x}^{n}{n}^{3}{a}^{2}-4\,ab{x}^{2}{n}^{2}-5\,a{n}^{2}xy-4\,{n}^{2}{a}^{2}{x}^{n}+6\,abn{x}^{2}+c{x}^{2}{n}^{2}+9\,anxy+6\,{a}^{2}{x}^{n}n-3\,ab{x}^{2}-3\,cn{x}^{2}-6\,axy-3\,{a}^{2}{x}^{n}+3\,c{x}^{2}}{ \left ( 2\,n{x}^{2}b+2\,nxy+2\,{x}^{n}an-3\,b{x}^{2}-3\,yx-3\,{x}^{n}a \right ) a}} \right ) -5\,\ln \left ( -9\,{\frac{ \left ( a{n}^{2}y-c{n}^{2}x-3\,any+3\,cnx+3\,ay-3\,cx \right ) x}{ \left ({x}^{n}a+b{x}^{2}+yx \right ) an}} \right ){n}^{2}+4\,{n}^{2}\ln \left ( 9\,{\frac{ab{x}^{2}{n}^{2}+{n}^{2}{a}^{2}{x}^{n}-3\,abn{x}^{2}+c{x}^{2}{n}^{2}-3\,{a}^{2}{x}^{n}n+3\,ab{x}^{2}-3\,cn{x}^{2}+3\,{a}^{2}{x}^{n}+3\,c{x}^{2}}{ \left ( n{x}^{2}b+nxy+{x}^{n}an-3\,b{x}^{2}-3\,yx-3\,{x}^{n}a \right ) a}} \right ) -3\,n\ln \left ( 9\,{\frac{{n}^{3}{x}^{2}ba+{n}^{3}xya+{x}^{n}{n}^{3}{a}^{2}-4\,ab{x}^{2}{n}^{2}-5\,a{n}^{2}xy-4\,{n}^{2}{a}^{2}{x}^{n}+6\,abn{x}^{2}+c{x}^{2}{n}^{2}+9\,anxy+6\,{a}^{2}{x}^{n}n-3\,ab{x}^{2}-3\,cn{x}^{2}-6\,axy-3\,{a}^{2}{x}^{n}+3\,c{x}^{2}}{ \left ( 2\,n{x}^{2}b+2\,nxy+2\,{x}^{n}an-3\,b{x}^{2}-3\,yx-3\,{x}^{n}a \right ) a}} \right ) +9\,\ln \left ( -9\,{\frac{ \left ( a{n}^{2}y-c{n}^{2}x-3\,any+3\,cnx+3\,ay-3\,cx \right ) x}{ \left ({x}^{n}a+b{x}^{2}+yx \right ) an}} \right ) n-6\,n\ln \left ( 9\,{\frac{ab{x}^{2}{n}^{2}+{n}^{2}{a}^{2}{x}^{n}-3\,abn{x}^{2}+c{x}^{2}{n}^{2}-3\,{a}^{2}{x}^{n}n+3\,ab{x}^{2}-3\,cn{x}^{2}+3\,{a}^{2}{x}^{n}+3\,c{x}^{2}}{ \left ( n{x}^{2}b+nxy+{x}^{n}an-3\,b{x}^{2}-3\,yx-3\,{x}^{n}a \right ) a}} \right ) +3\,\ln \left ( 9\,{\frac{{n}^{3}{x}^{2}ba+{n}^{3}xya+{x}^{n}{n}^{3}{a}^{2}-4\,ab{x}^{2}{n}^{2}-5\,a{n}^{2}xy-4\,{n}^{2}{a}^{2}{x}^{n}+6\,abn{x}^{2}+c{x}^{2}{n}^{2}+9\,anxy+6\,{a}^{2}{x}^{n}n-3\,ab{x}^{2}-3\,cn{x}^{2}-6\,axy-3\,{a}^{2}{x}^{n}+3\,c{x}^{2}}{ \left ( 2\,n{x}^{2}b+2\,nxy+2\,{x}^{n}an-3\,b{x}^{2}-3\,yx-3\,{x}^{n}a \right ) a}} \right ) -6\,\ln \left ( -9\,{\frac{ \left ( a{n}^{2}y-c{n}^{2}x-3\,any+3\,cnx+3\,ay-3\,cx \right ) x}{ \left ({x}^{n}a+b{x}^{2}+yx \right ) an}} \right ) +3\,\ln \left ( 9\,{\frac{ab{x}^{2}{n}^{2}+{n}^{2}{a}^{2}{x}^{n}-3\,abn{x}^{2}+c{x}^{2}{n}^{2}-3\,{a}^{2}{x}^{n}n+3\,ab{x}^{2}-3\,cn{x}^{2}+3\,{a}^{2}{x}^{n}+3\,c{x}^{2}}{ \left ( n{x}^{2}b+nxy+{x}^{n}an-3\,b{x}^{2}-3\,yx-3\,{x}^{n}a \right ) a}} \right ) \right ) }-6\,\ln \left ( x \right ) \right ) } \right ) \]

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39.43 problem number 43

problem number 312

Added January 2, 2019.

Problem 2.2.5.43 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a y^n+b x^2+c x y) w_x + \left (k y^n+ b x y+c y^2\right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(1,0)}(x,y) \left (a y^n+b x^2+c x y\right )+w^{(0,1)}(x,y) \left (b x y+c y^2+k y^n\right )=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( 1/3\,{\frac{1}{n-2} \left ( \ln \left ( ab{y}^{2}+ck{y}^{2}+{y}^{n}{k}^{2} \right ){n}^{2}-2\,{n}^{2}\ln \left ( y \right ) -3\,\ln \left ( ab{y}^{2}+ck{y}^{2}+{y}^{n}{k}^{2} \right ) n+6\,n\ln \left ( y \right ) +{\frac{n}{ \left ( n-2 \right ) \left ( n-1 \right ) } \left ( \ln \left ( -9\,{\frac{ \left ( -a{n}^{2}y+xk{n}^{2}+3\,any-3\,xkn-3\,ay+3\,kx \right ) by}{ \left ( k{y}^{n}+bxy+c{y}^{2} \right ) kn}} \right ){n}^{3}-{n}^{3}\ln \left ( 9\,{\frac{ab{n}^{2}{y}^{2}+ck{y}^{2}{n}^{2}+{y}^{n}{k}^{2}{n}^{2}-3\,abn{y}^{2}-3\,ckn{y}^{2}-3\,{y}^{n}{k}^{2}n+3\,ab{y}^{2}+3\,ck{y}^{2}+3\,{y}^{n}{k}^{2}}{ \left ( bnxy+cn{y}^{2}+k{y}^{n}n-3\,bxy-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) +{n}^{2}\ln \left ( 9\,{\frac{{n}^{3}ybxk+{n}^{3}{y}^{2}ck+{k}^{2}{y}^{n}{n}^{3}-5\,b{n}^{2}xyk+ab{n}^{2}{y}^{2}-4\,ck{y}^{2}{n}^{2}-4\,{y}^{n}{k}^{2}{n}^{2}+9\,bxykn-3\,abn{y}^{2}+6\,ckn{y}^{2}+6\,{y}^{n}{k}^{2}n-6\,bxyk+3\,ab{y}^{2}-3\,ck{y}^{2}-3\,{y}^{n}{k}^{2}}{ \left ( 2\,bnxy+2\,cn{y}^{2}+2\,k{y}^{n}n-3\,bxy-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) -5\,\ln \left ( -9\,{\frac{ \left ( -a{n}^{2}y+xk{n}^{2}+3\,any-3\,xkn-3\,ay+3\,kx \right ) by}{ \left ( k{y}^{n}+bxy+c{y}^{2} \right ) kn}} \right ){n}^{2}+4\,{n}^{2}\ln \left ( 9\,{\frac{ab{n}^{2}{y}^{2}+ck{y}^{2}{n}^{2}+{y}^{n}{k}^{2}{n}^{2}-3\,abn{y}^{2}-3\,ckn{y}^{2}-3\,{y}^{n}{k}^{2}n+3\,ab{y}^{2}+3\,ck{y}^{2}+3\,{y}^{n}{k}^{2}}{ \left ( bnxy+cn{y}^{2}+k{y}^{n}n-3\,bxy-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) -3\,n\ln \left ( 9\,{\frac{{n}^{3}ybxk+{n}^{3}{y}^{2}ck+{k}^{2}{y}^{n}{n}^{3}-5\,b{n}^{2}xyk+ab{n}^{2}{y}^{2}-4\,ck{y}^{2}{n}^{2}-4\,{y}^{n}{k}^{2}{n}^{2}+9\,bxykn-3\,abn{y}^{2}+6\,ckn{y}^{2}+6\,{y}^{n}{k}^{2}n-6\,bxyk+3\,ab{y}^{2}-3\,ck{y}^{2}-3\,{y}^{n}{k}^{2}}{ \left ( 2\,bnxy+2\,cn{y}^{2}+2\,k{y}^{n}n-3\,bxy-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) +9\,\ln \left ( -9\,{\frac{ \left ( -a{n}^{2}y+xk{n}^{2}+3\,any-3\,xkn-3\,ay+3\,kx \right ) by}{ \left ( k{y}^{n}+bxy+c{y}^{2} \right ) kn}} \right ) n-6\,n\ln \left ( 9\,{\frac{ab{n}^{2}{y}^{2}+ck{y}^{2}{n}^{2}+{y}^{n}{k}^{2}{n}^{2}-3\,abn{y}^{2}-3\,ckn{y}^{2}-3\,{y}^{n}{k}^{2}n+3\,ab{y}^{2}+3\,ck{y}^{2}+3\,{y}^{n}{k}^{2}}{ \left ( bnxy+cn{y}^{2}+k{y}^{n}n-3\,bxy-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) +3\,\ln \left ( 9\,{\frac{{n}^{3}ybxk+{n}^{3}{y}^{2}ck+{k}^{2}{y}^{n}{n}^{3}-5\,b{n}^{2}xyk+ab{n}^{2}{y}^{2}-4\,ck{y}^{2}{n}^{2}-4\,{y}^{n}{k}^{2}{n}^{2}+9\,bxykn-3\,abn{y}^{2}+6\,ckn{y}^{2}+6\,{y}^{n}{k}^{2}n-6\,bxyk+3\,ab{y}^{2}-3\,ck{y}^{2}-3\,{y}^{n}{k}^{2}}{ \left ( 2\,bnxy+2\,cn{y}^{2}+2\,k{y}^{n}n-3\,bxy-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) -6\,\ln \left ( -9\,{\frac{ \left ( -a{n}^{2}y+xk{n}^{2}+3\,any-3\,xkn-3\,ay+3\,kx \right ) by}{ \left ( k{y}^{n}+bxy+c{y}^{2} \right ) kn}} \right ) +3\,\ln \left ( 9\,{\frac{ab{n}^{2}{y}^{2}+ck{y}^{2}{n}^{2}+{y}^{n}{k}^{2}{n}^{2}-3\,abn{y}^{2}-3\,ckn{y}^{2}-3\,{y}^{n}{k}^{2}n+3\,ab{y}^{2}+3\,ck{y}^{2}+3\,{y}^{n}{k}^{2}}{ \left ( bnxy+cn{y}^{2}+k{y}^{n}n-3\,bxy-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) \right ) }+3\,\ln \left ( ab{y}^{2}+ck{y}^{2}+{y}^{n}{k}^{2} \right ) -6\,\ln \left ( y \right ) -2\,{\frac{1}{ \left ( n-2 \right ) \left ( n-1 \right ) } \left ( \ln \left ( -9\,{\frac{ \left ( -a{n}^{2}y+xk{n}^{2}+3\,any-3\,xkn-3\,ay+3\,kx \right ) by}{ \left ( k{y}^{n}+bxy+c{y}^{2} \right ) kn}} \right ){n}^{3}-{n}^{3}\ln \left ( 9\,{\frac{ab{n}^{2}{y}^{2}+ck{y}^{2}{n}^{2}+{y}^{n}{k}^{2}{n}^{2}-3\,abn{y}^{2}-3\,ckn{y}^{2}-3\,{y}^{n}{k}^{2}n+3\,ab{y}^{2}+3\,ck{y}^{2}+3\,{y}^{n}{k}^{2}}{ \left ( bnxy+cn{y}^{2}+k{y}^{n}n-3\,bxy-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) +{n}^{2}\ln \left ( 9\,{\frac{{n}^{3}ybxk+{n}^{3}{y}^{2}ck+{k}^{2}{y}^{n}{n}^{3}-5\,b{n}^{2}xyk+ab{n}^{2}{y}^{2}-4\,ck{y}^{2}{n}^{2}-4\,{y}^{n}{k}^{2}{n}^{2}+9\,bxykn-3\,abn{y}^{2}+6\,ckn{y}^{2}+6\,{y}^{n}{k}^{2}n-6\,bxyk+3\,ab{y}^{2}-3\,ck{y}^{2}-3\,{y}^{n}{k}^{2}}{ \left ( 2\,bnxy+2\,cn{y}^{2}+2\,k{y}^{n}n-3\,bxy-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) -5\,\ln \left ( -9\,{\frac{ \left ( -a{n}^{2}y+xk{n}^{2}+3\,any-3\,xkn-3\,ay+3\,kx \right ) by}{ \left ( k{y}^{n}+bxy+c{y}^{2} \right ) kn}} \right ){n}^{2}+4\,{n}^{2}\ln \left ( 9\,{\frac{ab{n}^{2}{y}^{2}+ck{y}^{2}{n}^{2}+{y}^{n}{k}^{2}{n}^{2}-3\,abn{y}^{2}-3\,ckn{y}^{2}-3\,{y}^{n}{k}^{2}n+3\,ab{y}^{2}+3\,ck{y}^{2}+3\,{y}^{n}{k}^{2}}{ \left ( bnxy+cn{y}^{2}+k{y}^{n}n-3\,bxy-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) -3\,n\ln \left ( 9\,{\frac{{n}^{3}ybxk+{n}^{3}{y}^{2}ck+{k}^{2}{y}^{n}{n}^{3}-5\,b{n}^{2}xyk+ab{n}^{2}{y}^{2}-4\,ck{y}^{2}{n}^{2}-4\,{y}^{n}{k}^{2}{n}^{2}+9\,bxykn-3\,abn{y}^{2}+6\,ckn{y}^{2}+6\,{y}^{n}{k}^{2}n-6\,bxyk+3\,ab{y}^{2}-3\,ck{y}^{2}-3\,{y}^{n}{k}^{2}}{ \left ( 2\,bnxy+2\,cn{y}^{2}+2\,k{y}^{n}n-3\,bxy-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) +9\,\ln \left ( -9\,{\frac{ \left ( -a{n}^{2}y+xk{n}^{2}+3\,any-3\,xkn-3\,ay+3\,kx \right ) by}{ \left ( k{y}^{n}+bxy+c{y}^{2} \right ) kn}} \right ) n-6\,n\ln \left ( 9\,{\frac{ab{n}^{2}{y}^{2}+ck{y}^{2}{n}^{2}+{y}^{n}{k}^{2}{n}^{2}-3\,abn{y}^{2}-3\,ckn{y}^{2}-3\,{y}^{n}{k}^{2}n+3\,ab{y}^{2}+3\,ck{y}^{2}+3\,{y}^{n}{k}^{2}}{ \left ( bnxy+cn{y}^{2}+k{y}^{n}n-3\,bxy-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) +3\,\ln \left ( 9\,{\frac{{n}^{3}ybxk+{n}^{3}{y}^{2}ck+{k}^{2}{y}^{n}{n}^{3}-5\,b{n}^{2}xyk+ab{n}^{2}{y}^{2}-4\,ck{y}^{2}{n}^{2}-4\,{y}^{n}{k}^{2}{n}^{2}+9\,bxykn-3\,abn{y}^{2}+6\,ckn{y}^{2}+6\,{y}^{n}{k}^{2}n-6\,bxyk+3\,ab{y}^{2}-3\,ck{y}^{2}-3\,{y}^{n}{k}^{2}}{ \left ( 2\,bnxy+2\,cn{y}^{2}+2\,k{y}^{n}n-3\,bxy-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) -6\,\ln \left ( -9\,{\frac{ \left ( -a{n}^{2}y+xk{n}^{2}+3\,any-3\,xkn-3\,ay+3\,kx \right ) by}{ \left ( k{y}^{n}+bxy+c{y}^{2} \right ) kn}} \right ) +3\,\ln \left ( 9\,{\frac{ab{n}^{2}{y}^{2}+ck{y}^{2}{n}^{2}+{y}^{n}{k}^{2}{n}^{2}-3\,abn{y}^{2}-3\,ckn{y}^{2}-3\,{y}^{n}{k}^{2}n+3\,ab{y}^{2}+3\,ck{y}^{2}+3\,{y}^{n}{k}^{2}}{ \left ( bnxy+cn{y}^{2}+k{y}^{n}n-3\,bxy-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) \right ) } \right ) } \right ) \]

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39.44 problem number 44

problem number 313

Added January 2, 2019.

Problem 2.2.5.44 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a x^n + b x^m + c) w_x + \left (c y^2-b x^{m-1} y+ a x^{n-2}\right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (a x^{n-2}-b y x^{m-1}+c y^2\right )+w^{(1,0)}(x,y) \left (a x^n+b x^m+c\right )=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ \text{ sol=() } \]

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39.45 problem number 45

problem number 314

Added January 2, 2019.

Problem 2.2.5.45 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a x^n + b x^m + c) w_x + \left (a x^{n-2} y^2 + b x^{m-1} y + c \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (a y^2 x^{n-2}+b y x^{m-1}+c\right )+w^{(1,0)}(x,y) \left (a x^n+b x^m+c\right )=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ \text{ sol=() } \]

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39.46 problem number 46

problem number 315

Added January 2, 2019.

Problem 2.2.5.46 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a x^n + b x^m + c) w_x + \left ( \alpha x^k y^2 + \beta x^s y - \alpha \lambda ^2 x^k + \beta \lambda x^s \right ) w_y = 0 \]

Mathematica

\[ \text{Timed out} \] Timed out

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{1 \left ( \int \!{\frac{\alpha \,{x}^{k}}{{x}^{n}a+{x}^{m}b+c}{{\rm e}^{\int \!{\frac{-2\,{x}^{k}\alpha \,\lambda +{x}^{s}\beta }{{x}^{n}a+{x}^{m}b+c}}\,{\rm d}x}}}\,{\rm d}x\lambda \,{{\rm e}^{\int \!{\frac{-2\,{x}^{k}\alpha \,\lambda +{x}^{s}\beta }{{x}^{n}a+{x}^{m}b+c}}\,{\rm d}x+\int \!-{\frac{-2\,{x}^{k}\alpha \,\lambda +{x}^{s}\beta }{{x}^{n}a+{x}^{m}b+c}}\,{\rm d}x}}+y\int \!{\frac{\alpha \,{x}^{k}}{{x}^{n}a+{x}^{m}b+c}{{\rm e}^{\int \!{\frac{-2\,{x}^{k}\alpha \,\lambda +{x}^{s}\beta }{{x}^{n}a+{x}^{m}b+c}}\,{\rm d}x}}}\,{\rm d}x+{{\rm e}^{\int \!{\frac{-2\,{x}^{k}\alpha \,\lambda +{x}^{s}\beta }{{x}^{n}a+{x}^{m}b+c}}\,{\rm d}x}} \right ) \left ({{\rm e}^{\int \!{\frac{-2\,{x}^{k}\alpha \,\lambda +{x}^{s}\beta }{{x}^{n}a+{x}^{m}b+c}}\,{\rm d}x+\int \!-{\frac{-2\,{x}^{k}\alpha \,\lambda +{x}^{s}\beta }{{x}^{n}a+{x}^{m}b+c}}\,{\rm d}x}}\lambda +y \right ) ^{-1}} \right ) \]

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39.47 problem number 47

problem number 316

Added January 2, 2019.

Problem 2.2.5.47 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x(a x^n + b x^m + c) w_x - \left ( s x^k y^2 -(a x^n + b x^m+c) y - s \lambda x^{k+2} \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{\tanh ^{-1}\left (\frac{y}{\sqrt{\lambda } x}\right )-\sqrt{\lambda } \int _1^x \frac{s K[1]^k}{a K[1]^n+b K[1]^m+c} \, dK[1]}{\sqrt{\lambda }}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{s\sqrt{\lambda }} \left ( -\int \!{\frac{{x}^{k}}{{x}^{n}a+{x}^{m}b+c}}\,{\rm d}xs\sqrt{\lambda }+\arctanh \left ({\frac{y}{x\sqrt{\lambda }}} \right ) \right ) } \right ) \]

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39.48 problem number 48

problem number 317

Added January 2, 2019.

Problem 2.2.5.48 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a x^n + b x^m + c) w_x + \left ( (a x^n+b x^m + c)y^2-a n(n-1)x^{n-2}-b m(m-1) x^{m-2}\right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(1,0)}(x,y) \left (a x^n+b x^m+c\right )+w^{(0,1)}(x,y) \left (y^2 \left (a x^n+b x^m+c\right )-a (n-1) n x^{n-2}-b (m-1) m x^{m-2}\right )=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{1}{2\,{x}^{m+n+1}aby+2\,ay{x}^{n+1}c+2\,cb{x}^{m+1}y+{x}^{n+m}abm+{x}^{n+m}abn+{x}^{m}bcm+y{x}^{1+2\,n}{a}^{2}+{x}^{2\,m+1}{b}^{2}y+{x}^{2\,n}n{a}^{2}+{x}^{2\,m}{b}^{2}m+{c}^{2}xy+{x}^{n}acn} \left ( x+2\,y \left ( -{\frac{x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ({x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac{{m}^{2}{x}^{m}b-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ({x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ){x}^{m+n+1}ab+2\,y \left ( -{\frac{x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ({x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac{{m}^{2}{x}^{m}b-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ({x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ){x}^{n+1}ac+2\,y \left ( -{\frac{x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ({x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac{{m}^{2}{x}^{m}b-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ({x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ){x}^{m+1}bc+y \left ( -{\frac{x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ({x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac{{m}^{2}{x}^{m}b-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ({x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ){x}^{1+2\,n}{a}^{2}+y \left ( -{\frac{x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ({x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac{{m}^{2}{x}^{m}b-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ({x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ){x}^{2\,m+1}{b}^{2}+ \left ( -{\frac{x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ({x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac{{m}^{2}{x}^{m}b-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ({x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ){x}^{2\,n}{a}^{2}n+ \left ( -{\frac{x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ({x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac{{m}^{2}{x}^{m}b-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ({x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ){x}^{2\,m}{b}^{2}m+ \left ( -{\frac{x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ({x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac{{m}^{2}{x}^{m}b-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ({x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ){x}^{n}acn+ \left ( -{\frac{x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ({x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac{{m}^{2}{x}^{m}b-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ({x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ){x}^{m}bcm+y \left ( -{\frac{x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ({x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac{{m}^{2}{x}^{m}b-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ({x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ){c}^{2}x+ \left ( -{\frac{x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ({x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac{{m}^{2}{x}^{m}b-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ({x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ){x}^{n+m}abm+ \left ( -{\frac{x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ({x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac{{m}^{2}{x}^{m}b-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ({x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ){x}^{n+m}abn \right ) } \right ) \]

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39.49 problem number 49

problem number 318

Added January 2, 2019.

Problem 2.2.5.49 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a x^n + b y^n + x) w_x + \left ( \alpha x^k y^{n-k} + \beta x^m y^{n-m} + y \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(1,0)}(x,y) \left (a x^n+b y^n+x\right )+w^{(0,1)}(x,y) \left (\alpha x^k y^{n-k}+\beta x^m y^{n-m}+y\right )=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ \text{ sol=() } \]

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39.50 problem number 50

problem number 319

Added January 2, 2019.

Problem 2.2.5.50 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a x^n + b y^n + A x^2 + B x y) w_x + \left ( \alpha x^k y^{n-k} + \beta x^m y^{n-m} + A x y + B y^2\right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(1,0)}(x,y) \left (a x^n+A x^2+b y^n+B x y\right )+w^{(0,1)}(x,y) \left (A x y+\alpha x^k y^{n-k}+B y^2+\beta x^m y^{n-m}\right )=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ \text{ sol=() } \]

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39.51 problem number 51

problem number 320

Added January 2, 2019.

Problem 2.2.5.51 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a y^m + b x^n + s) w_x - \left ( \alpha x^k + b n x^{n-1} y + \beta \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(1,0)}(x,y) \left (a y^m+b x^n+s\right )-w^{(0,1)}(x,y) \left (\alpha x^k+b n y x^{n-1}+\beta \right )=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{{x}^{n}bkmy+{x}^{k}x\alpha \,m+{x}^{n}bky+bym{x}^{n}+\beta \,kmx+kmsy+a{y}^{m}yk+{x}^{k}\alpha \,x+{x}^{n}by+\beta \,kx+\beta \,mx+ksy+msy+a{y}^{m}y+\beta \,x+sy}{mk+k+m+1}} \right ) \]

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39.52 problem number 52

problem number 321

Added January 2, 2019.

Problem 2.2.5.52 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a x^n y^m +x) w_x + \left ( b x^k y^{n+m-k} + y \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(1,0)}(x,y) \left (a y^m x^n+x\right )+w^{(0,1)}(x,y) \left (b x^k y^{-k+m+n}+y\right )=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ \text{ sol=() } \]

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39.53 problem number 53

problem number 322

Added January 2, 2019.

Problem 2.2.5.53 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x(a x^n y^m +\alpha ) w_x - y \left ( b x^n y^m + \beta \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [x w^{(1,0)}(x,y) \left (a y^m x^n+\alpha \right )-y w^{(0,1)}(x,y) \left (b y^m x^n+\beta \right )=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( \left ({x}^{n}{y}^{m}an-b{x}^{n}{y}^{m}m+\alpha \,n-\beta \,m \right ) ^{m \left ( a\beta -\alpha \,b \right ) }{x}^{-\beta \,m \left ( an-bm \right ) } \left ({y}^{m} \right ) ^{-\alpha \, \left ( an-bm \right ) } \right ) \]

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39.54 problem number 54

problem number 323

Added January 2, 2019.

Problem 2.2.5.54 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x(a n x^k y^{n+k} + s) w_x - y \left ( b m x^{m+k} y^k + s \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [x w^{(1,0)}(x,y) \left (a n x^k y^{k+n}+s\right )-y w^{(0,1)}(x,y) \left (b m y^k x^{k+m}+s\right )=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ \text{ sol=() } \]

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39.55 problem number 55

problem number 324

Added January 2, 2019.

Problem 2.2.5.55 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a x^n y^m + A x^2 + B x y) w_x + \left ( b x^k y^{n+m-k} + A x y+ B y^2 \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(1,0)}(x,y) \left (a y^m x^n+A x^2+B x y\right )+w^{(0,1)}(x,y) \left (A x y+b x^k y^{-k+m+n}+B y^2\right )=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ \text{ sol=() } \]

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39.56 problem number 56

problem number 325

Added January 2, 2019.

Problem 2.2.5.56 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a x^n y^m + b x y^k) w_x + \left ( \alpha y^s + \beta \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(1,0)}(x,y) \left (a y^m x^n+b x y^k\right )+w^{(0,1)}(x,y) \left (\alpha y^s+\beta \right )=0,w(x,y),\{x,y\}\right ] \] Timed out

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({x}^{-n+1}{{\rm e}^{b \left ( n-1 \right ) \int \!{\frac{{y}^{k}}{\alpha \,{y}^{s}+\beta }}\,{\rm d}y}}+an\int \!{\frac{{y}^{m}}{\alpha \,{y}^{s}+\beta }{{\rm e}^{b \left ( n-1 \right ) \int \!{\frac{{y}^{k}}{\alpha \,{y}^{s}+\beta }}\,{\rm d}y}}}\,{\rm d}y-a\int \!{\frac{{y}^{m}}{\alpha \,{y}^{s}+\beta }{{\rm e}^{b \left ( n-1 \right ) \int \!{\frac{{y}^{k}}{\alpha \,{y}^{s}+\beta }}\,{\rm d}y}}}\,{\rm d}y \right ) \]