41 HFOPDE, chapter 2.3.2

 41.1 problem number 1
 41.2 problem number 2
 41.3 problem number 3
 41.4 problem number 4
 41.5 problem number 5
 41.6 problem number 6
 41.7 problem number 7
 41.8 problem number 8
 41.9 problem number 9
 41.10 problem number 10
 41.11 problem number 11
 41.12 problem number 12
 41.13 problem number 13
 41.14 problem number 14
 41.15 problem number 15
 41.16 problem number 16
 41.17 problem number 17
 41.18 problem number 18
 41.19 problem number 19
 41.20 problem number 20
 41.21 problem number 21
 41.22 problem number 22
 41.23 problem number 23
 41.24 problem number 24
 41.25 problem number 25
 41.26 problem number 26
 41.27 problem number 27
 41.28 problem number 28
 41.29 problem number 29
 41.30 problem number 30
 41.31 problem number 31
 41.32 problem number 32
 41.33 problem number 33
 41.34 problem number 34
 41.35 problem number 35
 41.36 problem number 36

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41.1 problem number 1

problem number 337

Added January 7, 2019.

Problem 2.3.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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41.2 problem number 2

problem number 338

Added January 7, 2019.

Problem 2.3.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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41.3 problem number 3

problem number 339

Added January 7, 2019.

Problem 2.3.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{2 b (-1)^{-\frac{b}{\lambda }} \lambda ^{-\frac{2 b}{\lambda }-1} \left (y \lambda ^{\frac{2 b}{\lambda }} \text{Gamma}\left (\frac{2 b}{\lambda },0,-\frac{a e^{\lambda x}}{\lambda }\right )-b \lambda ^{\frac{2 b}{\lambda }} \text{Gamma}\left (\frac{2 b}{\lambda },0,-\frac{a e^{\lambda x}}{\lambda }\right )+\lambda (-1)^{\frac{2 b}{\lambda }} a^{\frac{2 b}{\lambda }} e^{\frac{a e^{\lambda x}}{\lambda }+2 b x}\right )}{b-y}\right )\right \}\right \} \]

Maple

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

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41.4 problem number 4

problem number 340

Added January 7, 2019.

Problem 2.3.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{x{\lambda }^{2} \left ( yx-1 \right ) } \left ( y{x}^{2}\int \!{\frac{1}{{x}^{2}}{{\rm e}^{{\frac{{{\rm e}^{\lambda \,x}}a \left ( \lambda \,x-1 \right ) }{{\lambda }^{2}}}}}}\,{\rm d}x-{{\rm e}^{{\frac{{{\rm e}^{\lambda \,x}}a \left ( \lambda \,x-1 \right ) }{{\lambda }^{2}}}}}-x\int \!{\frac{1}{{x}^{2}}{{\rm e}^{{\frac{{{\rm e}^{\lambda \,x}}a \left ( \lambda \,x-1 \right ) }{{\lambda }^{2}}}}}}\,{\rm d}x \right ) } \right ) \]

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41.5 problem number 5

problem number 341

Added January 7, 2019.

Problem 2.3.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{\lambda }{\sqrt{{\lambda }^{2} \left ( 4\,ab-{\lambda }^{2} \right ) }} \left ( 2\,\lambda \,\arctan \left ({\frac{\lambda \, \left ( 2\,{{\rm e}^{\lambda \,x}}ay+\lambda \right ) }{\sqrt{4\,{\lambda }^{2}ab-{\lambda }^{4}}}} \right ) -\sqrt{{\lambda }^{2} \left ( 4\,ab-{\lambda }^{2} \right ) }x \right ) } \right ) \]

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41.6 problem number 6

problem number 342

Added January 7, 2019.

Problem 2.3.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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41.7 problem number 7

problem number 343

Added January 7, 2019.

Problem 2.3.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{\sqrt{ \left ( b+\lambda \right ) ^{2} \left ( 4\,ca-{b}^{2}-2\,\lambda \,b-{\lambda }^{2} \right ) }} \left ( 2\,{b}^{2}\arctan \left ({\frac{2\,y{{\rm e}^{\lambda \,x}}ab+2\,a\lambda \,{{\rm e}^{\lambda \,x}}y+{b}^{2}+2\,\lambda \,b+{\lambda }^{2}}{\sqrt{ \left ( b+\lambda \right ) ^{2} \left ( 4\,ca-{b}^{2}-2\,\lambda \,b-{\lambda }^{2} \right ) }}} \right ) +4\,b\lambda \,\arctan \left ({\frac{2\,y{{\rm e}^{\lambda \,x}}ab+2\,a\lambda \,{{\rm e}^{\lambda \,x}}y+{b}^{2}+2\,\lambda \,b+{\lambda }^{2}}{\sqrt{ \left ( b+\lambda \right ) ^{2} \left ( 4\,ca-{b}^{2}-2\,\lambda \,b-{\lambda }^{2} \right ) }}} \right ) +2\,{\lambda }^{2}\arctan \left ({\frac{2\,y{{\rm e}^{\lambda \,x}}ab+2\,a\lambda \,{{\rm e}^{\lambda \,x}}y+{b}^{2}+2\,\lambda \,b+{\lambda }^{2}}{\sqrt{ \left ( b+\lambda \right ) ^{2} \left ( 4\,ca-{b}^{2}-2\,\lambda \,b-{\lambda }^{2} \right ) }}} \right ) -\sqrt{ \left ( b+\lambda \right ) ^{2} \left ( 4\,ca-{b}^{2}-2\,\lambda \,b-{\lambda }^{2} \right ) }bx-\sqrt{ \left ( b+\lambda \right ) ^{2} \left ( 4\,ca-{b}^{2}-2\,\lambda \,b-{\lambda }^{2} \right ) }\lambda \,x \right ) } \right ) \]

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41.8 problem number 8

problem number 344

Added January 7, 2019.

Problem 2.3.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (-a b^2 e^{x (\lambda +2 \mu )}+a y^2 e^{\lambda x}+\mu y\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \] kernel error generated

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{1 \left ( y\sinh \left ({\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{\lambda \,x}}+{{\rm e}^{x \left ( \lambda +\mu \right ) }}\cosh \left ({\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) b \right ) \left ( y\cosh \left ({\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{\lambda \,x}}+{{\rm e}^{x \left ( \lambda +\mu \right ) }}\sinh \left ({\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) b \right ) ^{-1}} \right ) \]

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41.9 problem number 9

problem number 345

Added January 7, 2019.

Problem 2.3.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{{{\rm e}^{\lambda \,x}} \left ( y{{\rm e}^{\lambda \,x}}\lambda -y{{\rm e}^{\lambda \,x}}\mu +{\lambda }^{2}-\lambda \,\mu \right ) \left ({{\rm e}^{\mu \,x}}{\mbox{$_1$F$_1$}(-{\frac{\lambda -\mu }{\mu }};\,-{\frac{\lambda -2\,\mu }{\mu }};\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }})}a\lambda +y{{\rm e}^{\lambda \,x}}{\mbox{$_1$F$_1$}(-{\frac{\lambda }{\mu }};\,-{\frac{\lambda -\mu }{\mu }};\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }})}\lambda -y{{\rm e}^{\lambda \,x}}{\mbox{$_1$F$_1$}(-{\frac{\lambda }{\mu }};\,-{\frac{\lambda -\mu }{\mu }};\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }})}\mu \right ) ^{-1}} \right ) \]

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41.10 problem number 10

problem number 346

Added January 7, 2019.

Problem 2.3.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{1 \left ( -y{{\rm e}^{\lambda \,x}}{{\sl M}\left (-{\frac{\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\lambda +{{\rm e}^{\mu \,x}}{{\sl M}\left (-{\frac{\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}a-{{\sl M}\left (-{\frac{\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}{\lambda }^{2}+{{\sl M}\left (-{\frac{{\lambda }^{2}-\lambda +\mu }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}{\lambda }^{2}+{{\sl M}\left (-{\frac{\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\lambda -{{\sl M}\left (-{\frac{\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\mu +{{\sl M}\left (-{\frac{{\lambda }^{2}-\lambda +\mu }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\mu \right ) \left ( -y{{\rm e}^{\lambda \,x}}{{\sl U}\left (-{\frac{\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\lambda +{{\rm e}^{\mu \,x}}{{\sl U}\left (-{\frac{\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}a-{{\sl U}\left (-{\frac{\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}{\lambda }^{2}+{{\sl U}\left (-{\frac{\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\lambda -{{\sl U}\left (-{\frac{\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\mu -{{\sl U}\left (-{\frac{{\lambda }^{2}-\lambda +\mu }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\mu \right ) ^{-1}} \right ) \]

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41.11 problem number 11

problem number 347

Added January 7, 2019.

Problem 2.3.2.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{a \left ({{\rm e}^{\lambda \,x}}y+{{\rm e}^{x \left ( \lambda +\mu \right ) }}b \right ){{\rm e}^{{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }}}} \left ( -4\, \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}{\lambda }^{3}- \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}{\mu }^{3}+12\, \WhittakerM \left ( 1/2\,{\frac{4\,\lambda +3\,\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}{\lambda }^{3}+2\, \WhittakerM \left ( 1/2\,{\frac{4\,\lambda +3\,\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}{\mu }^{3}-2\,{{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}} \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){\lambda }^{3}-{{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}} \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){\mu }^{3}+y \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) a{\mu }^{2}{{\rm e}^{\lambda \,x+1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+2\,y \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) a{\lambda }^{2}{{\rm e}^{\lambda \,x+1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+y \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) a{\mu }^{2}{{\rm e}^{\lambda \,x+1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+ \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ab{\mu }^{2}{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+{a}^{2}{b}^{2} \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) \lambda \,{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-\mu \,x\lambda -{\mu }^{2}x}{\lambda +\mu }}}}+{a}^{2}{b}^{2} \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) \mu \,{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-\mu \,x\lambda -{\mu }^{2}x}{\lambda +\mu }}}}+4\, \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ab{\lambda }^{2}{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+ \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ab{\mu }^{2}{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+2\, \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ab{\lambda }^{2}{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+2\, \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-\mu \,x\lambda -{\mu }^{2}x}{\lambda +\mu }}}}ab{\lambda }^{2}+ \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-\mu \,x\lambda -{\mu }^{2}x}{\lambda +\mu }}}}ab{\mu }^{2}-5\, \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}\lambda \,{\mu }^{2}+20\, \WhittakerM \left ( 1/2\,{\frac{4\,\lambda +3\,\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}{\lambda }^{2}\mu +11\, \WhittakerM \left ( 1/2\,{\frac{4\,\lambda +3\,\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}\lambda \,{\mu }^{2}-5\,{{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}} \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){\lambda }^{2}\mu -4\,{{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}} \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) \lambda \,{\mu }^{2}-8\, \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}{\lambda }^{2}\mu +4\,y \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) a{\lambda }^{2}{{\rm e}^{\lambda \,x+1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+4\,y \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) a\lambda \,\mu \,{{\rm e}^{\lambda \,x+1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+3\,y \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) a\lambda \,\mu \,{{\rm e}^{\lambda \,x+1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+4\, \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ab\lambda \,\mu \,{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+3\, \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ab\lambda \,\mu \,{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+y \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){a}^{2}b\lambda \,{{\rm e}^{\lambda \,x+1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-\mu \,x\lambda -{\mu }^{2}x}{\lambda +\mu }}}}+y \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){a}^{2}b\mu \,{{\rm e}^{\lambda \,x+1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-\mu \,x\lambda -{\mu }^{2}x}{\lambda +\mu }}}}+3\, \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-\mu \,x\lambda -{\mu }^{2}x}{\lambda +\mu }}}}ab\lambda \,\mu \right ) ^{-1}} \right ) \]

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41.12 problem number 12

problem number 348

Added January 7, 2019.

Problem 2.3.2.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{i \pi e^{-\frac{\sqrt{b^2-4 a c} e^{x (\lambda +\mu )}}{2 (\lambda +\mu )}} \left (\sqrt{b^2-4 a c} e^{x (\lambda +\mu )}-2 a y e^{x (2 \lambda +\mu )}+b \left (-e^{x (\lambda +\mu )}\right )\right )}{2 \left (2 a y e^{x (2 \lambda +\mu )} \cosh \left (\frac{\sqrt{b^2-4 a c} e^{x (\lambda +\mu )}}{2 (\lambda +\mu )}\right )+\sqrt{b^2-4 a c} e^{x (\lambda +\mu )} \sinh \left (\frac{\sqrt{b^2-4 a c} e^{x (\lambda +\mu )}}{2 (\lambda +\mu )}\right )+b e^{x (\lambda +\mu )} \cosh \left (\frac{\sqrt{b^2-4 a c} e^{x (\lambda +\mu )}}{2 (\lambda +\mu )}\right )\right )}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{b}{\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) } \left ( \lambda +\mu \right ) } \left ( 2\,b\lambda \,\arctan \left ({\frac{b \left ( 2\,{{\rm e}^{\lambda \,x}}ay+b \right ) }{\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }}} \right ) +2\,b\mu \,\arctan \left ({\frac{b \left ( 2\,{{\rm e}^{\lambda \,x}}ay+b \right ) }{\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }}} \right ) -{{\rm e}^{x \left ( \lambda +\mu \right ) }}\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) } \right ) } \right ) \]

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41.13 problem number 13

problem number 349

Added January 7, 2019.

Problem 2.3.2.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{b e^{\lambda x+\mu x}-y e^{\lambda x}-\lambda }{\lambda \left (b e^{\mu x}-y\right )}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{b{{\rm e}^{\lambda \,x+\mu \,x}}-{{\rm e}^{\lambda \,x}}y-\lambda }{\lambda \, \left ( b{{\rm e}^{\mu \,x}}-y \right ) }} \right ) \]

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41.14 problem number 14

problem number 350

Added January 7, 2019.

Problem 2.3.2.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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41.15 problem number 15

problem number 351

Added January 7, 2019.

Problem 2.3.2.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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41.16 problem number 16

problem number 352

Added January 7, 2019.

Problem 2.3.2.16 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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41.17 problem number 17

problem number 353

Added January 7, 2019.

Problem 2.3.2.17 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{a \left ( -b{x}^{n}+y \right ) \left ( -{x}^{n}\int \!\lambda \,{{\rm e}^{-{\frac{ab \left ( -\lambda \right ) ^{-n} \left ({x}^{n} \left ( -\lambda \right ) ^{n}n\Gamma \left ( n \right ) \left ( -\lambda \,x \right ) ^{-n}-{x}^{n} \left ( -\lambda \right ) ^{n}{{\rm e}^{\lambda \,x}}-{x}^{n} \left ( -\lambda \right ) ^{n}n \left ( -\lambda \,x \right ) ^{-n}\Gamma \left ( n,-\lambda \,x \right ) \right ) }{\lambda }}+\lambda \,x}}\,{\rm d}xab+\int \!\lambda \,{{\rm e}^{-{\frac{ab \left ( -\lambda \right ) ^{-n} \left ({x}^{n} \left ( -\lambda \right ) ^{n}n\Gamma \left ( n \right ) \left ( -\lambda \,x \right ) ^{-n}-{x}^{n} \left ( -\lambda \right ) ^{n}{{\rm e}^{\lambda \,x}}-{x}^{n} \left ( -\lambda \right ) ^{n}n \left ( -\lambda \,x \right ) ^{-n}\Gamma \left ( n,-\lambda \,x \right ) \right ) }{\lambda }}+\lambda \,x}}\,{\rm d}xya+\lambda \,{{\rm e}^{-{\frac{ab \left ( -\lambda \right ) ^{-n} \left ({x}^{n} \left ( -\lambda \right ) ^{n}n\Gamma \left ( n \right ) \left ( -\lambda \,x \right ) ^{-n}-{x}^{n} \left ( -\lambda \right ) ^{n}{{\rm e}^{\lambda \,x}}-{x}^{n} \left ( -\lambda \right ) ^{n}n \left ( -\lambda \,x \right ) ^{-n}\Gamma \left ( n,-\lambda \,x \right ) \right ) }{\lambda }}}} \right ) ^{-1}} \right ) \]

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41.18 problem number 18

problem number 354

Added January 7, 2019.

Problem 2.3.2.18 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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41.19 problem number 19

problem number 355

Added January 7, 2019.

Problem 2.3.2.19 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{-i}{\lambda } \left ( b{x}^{n}\Gamma \left ( n,-\lambda \,x \right ) n \left ( -\lambda \,x \right ) ^{-n}a-bn{x}^{n} \left ( -\lambda \,x \right ) ^{-n}a\Gamma \left ( n \right ) +ab{x}^{n}{{\rm e}^{\lambda \,x}}+\arctanh \left ({\frac{{{\rm e}^{-\lambda \,x}}y}{b}} \right ) \lambda \right ) } \right ) \]

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41.20 problem number 20

problem number 356

Added January 7, 2019.

Problem 2.3.2.20 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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41.21 problem number 21

problem number 357

Added January 7, 2019.

Problem 2.3.2.21 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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41.22 problem number 22

problem number 358

Added January 7, 2019.

Problem 2.3.2.22 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{-4 a^{3/2} c^{3/2} (-1)^{-n} \lambda ^{-n-1} \text{Gamma}(n+1,-\lambda x)+\sqrt{a} b^2 \sqrt{c} (-1)^{-n} \lambda ^{-n-1} \text{Gamma}(n+1,-\lambda x)+2 \sqrt{a} \sqrt{c} \sqrt{4 a c-b^2} \tan ^{-1}\left (\frac{2 a y e^{\lambda x} \sqrt{4 a c-b^2}-b \sqrt{4 a c-b^2}}{4 a c-b^2}\right )}{4 a c-b^2}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{b}{\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }\lambda } \left ( - \left ( -\lambda \,x \right ) ^{-n}\Gamma \left ( n,-\lambda \,x \right ) \sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{x}^{n}n+ \left ( -\lambda \,x \right ) ^{-n}\Gamma \left ( n \right ) \sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{x}^{n}n-{{\rm e}^{\lambda \,x}}\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{x}^{n}+2\,b\lambda \,\arctan \left ({\frac{b \left ( 2\,{{\rm e}^{\lambda \,x}}ay+b \right ) }{\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }}} \right ) \right ) } \right ) \]

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41.23 problem number 23

problem number 359

Added January 10, 2019.

Problem 2.3.2.23 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{-ab{x}^{n}{{\rm e}^{\lambda \,x}}+{{\rm e}^{\lambda \,x}}ay-{{\rm e}^{\lambda \,x}}ac+\lambda }{\lambda \, \left ( y-b{x}^{n}-c \right ) }} \right ) \]

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41.24 problem number 24

problem number 360

Added January 10, 2019.

Problem 2.3.2.24 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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41.25 problem number 25

problem number 361

Added January 10, 2019.

Problem 2.3.2.25 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{1}{2} \left (2 \tan ^{-1}\left (\frac{y e^{-\frac{\lambda x^2}{2}}}{b}\right )-\frac{\sqrt{2 \pi } a b \text{Erf}\left (\frac{\sqrt{\lambda } x}{\sqrt{2}}\right )}{\sqrt{\lambda }}\right )\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( 1/2\,{\frac{1}{\sqrt{\lambda }} \left ( ab\sqrt{\pi }\sqrt{2}\erf \left ( 1/2\,\sqrt{2}\sqrt{\lambda }x \right ) -2\,\arctan \left ({\frac{{{\rm e}^{-1/2\,{x}^{2}\lambda }}y}{b}} \right ) \sqrt{\lambda } \right ) } \right ) \]

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41.26 problem number 26

problem number 362

Added January 10, 2019.

Problem 2.3.2.26 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({2}^{n/2-1/2}{x}^{n+1}ab \left ( -{x}^{2}\lambda \right ) ^{-1/2-n/2}\Gamma \left ( n/2+1/2 \right ) -{2}^{n/2-1/2}{x}^{n+1}ab \left ( -{x}^{2}\lambda \right ) ^{-1/2-n/2}\Gamma \left ( n/2+1/2,-1/2\,{x}^{2}\lambda \right ) -\arctan \left ({\frac{{{\rm e}^{-1/2\,{x}^{2}\lambda }}y}{b}} \right ) \right ) \]

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41.27 problem number 27

problem number 363

Added January 10, 2019.

Problem 2.3.2.27 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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41.28 problem number 28

problem number 364

Added January 10, 2019.

Problem 2.3.2.28 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{e^{-\frac{6 a b^2 e^{\lambda x}}{\lambda }} \left (2 y^2 e^{\frac{6 a b^2 e^{\lambda x}}{\lambda }} \int _1^x a \exp \left (-\frac{6 a b^2 e^{\lambda K[1]}}{\lambda }+2 c K[1]+\lambda K[1]\right ) \, dK[1]+4 b y e^{\frac{6 a b^2 e^{\lambda x}}{\lambda }} \left (\int _1^x a \exp \left (-\frac{6 a b^2 e^{\lambda K[1]}}{\lambda }+2 c K[1]+\lambda K[1]\right ) \, dK[1]\right )+2 b^2 e^{\frac{6 a b^2 e^{\lambda x}}{\lambda }} \left (\int _1^x a \exp \left (-\frac{6 a b^2 e^{\lambda K[1]}}{\lambda }+2 c K[1]+\lambda K[1]\right ) \, dK[1]\right )+e^{2 c x}\right )}{(b+y)^2}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{ \left ( b+y \right ) ^{2}} \left ( 2\,{b}^{2}a\int \!{{\rm e}^{-{\frac{6\,a{{\rm e}^{\lambda \,x}}{b}^{2}-2\,c\lambda \,x-x{\lambda }^{2}}{\lambda }}}}\,{\rm d}x+4\,yab\int \!{{\rm e}^{-{\frac{6\,a{{\rm e}^{\lambda \,x}}{b}^{2}-2\,c\lambda \,x-x{\lambda }^{2}}{\lambda }}}}\,{\rm d}x+2\,{y}^{2}a\int \!{{\rm e}^{-{\frac{6\,a{{\rm e}^{\lambda \,x}}{b}^{2}-2\,c\lambda \,x-x{\lambda }^{2}}{\lambda }}}}\,{\rm d}x+{{\rm e}^{2\,cx-6\,{\frac{a{{\rm e}^{\lambda \,x}}{b}^{2}}{\lambda }}}} \right ) } \right ) \]

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41.29 problem number 29

problem number 365

Added January 10, 2019.

Problem 2.3.2.29 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (a \sqrt{b^2} x^k (-\lambda x)^{-k} \text{Gamma}(k,-\lambda x)+\tan ^{-1}\left (\frac{y x^{-k}}{\sqrt{b^2}}\right )\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( b{x}^{k} \left ( -\lambda \,x \right ) ^{-k}a\Gamma \left ( k \right ) -b{x}^{k} \left ( -\lambda \,x \right ) ^{-k}\Gamma \left ( k,-\lambda \,x \right ) a-\arctan \left ({\frac{{x}^{-k}y}{b}} \right ) \right ) \]

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41.30 problem number 30

problem number 366

Added January 10, 2019.

Problem 2.3.2.30 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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41.31 problem number 31

problem number 367

Added January 10, 2019.

Problem 2.3.2.31 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac{1}{a} \left ( 2\,\lambda \,xa{{\rm e}^{2\,{1\arctan \left ({\frac{2\,y\lambda \,{{\rm e}^{-\lambda \,x}}-2\,\lambda \,xa-b}{a}{\frac{1}{\sqrt{-{\frac{{b}^{2}+4\,\lambda \,c}{{a}^{2}}}}}}} \right ){\frac{1}{\sqrt{-{\frac{{b}^{2}+4\,\lambda \,c}{{a}^{2}}}}}}}}}+\sqrt{-{\frac{{b}^{2}+4\,\lambda \,c}{{a}^{2}}}}\int ^{-2\,{1\arctan \left ({\frac{2\,y\lambda \,{{\rm e}^{-\lambda \,x}}-2\,\lambda \,xa-b}{a}{\frac{1}{\sqrt{-{\frac{{b}^{2}+4\,\lambda \,c}{{a}^{2}}}}}}} \right ){\frac{1}{\sqrt{-{\frac{{b}^{2}+4\,\lambda \,c}{{a}^{2}}}}}}}}\!\tan \left ( 1/2\,{\it \_a}\,\sqrt{-{\frac{{b}^{2}+4\,\lambda \,c}{{a}^{2}}}} \right ){{\rm e}^{-{\it \_a}}}{d{\it \_a}}a+b{{\rm e}^{2\,{1\arctan \left ({\frac{2\,y\lambda \,{{\rm e}^{-\lambda \,x}}-2\,\lambda \,xa-b}{a}{\frac{1}{\sqrt{-{\frac{{b}^{2}+4\,\lambda \,c}{{a}^{2}}}}}}} \right ){\frac{1}{\sqrt{-{\frac{{b}^{2}+4\,\lambda \,c}{{a}^{2}}}}}}}}} \right ) } \right ) \]

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41.32 problem number 32

problem number 368

Added January 10, 2019.

Problem 2.3.2.32 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a e^{\lambda x} w_x + b y^m w_y = 0 \]

Mathematica

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

Maple

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

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41.33 problem number 33

problem number 369

Added January 10, 2019.

Problem 2.3.2.33 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a e^y + b x) w_x + w_y = 0 \]

Mathematica

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

Maple

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

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41.34 problem number 34

problem number 370

Added January 10, 2019.

Problem 2.3.2.34 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x^n e^{\lambda y} + b x y^m) w_x + e^{\mu y} w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({{x}^{ \left ( m+1 \right ) ^{-1}}{{\rm e}^{{\frac{bn{{\rm e}^{-1/2\,\mu \,y}}{y}^{m} \left ( \mu \,y \right ) ^{-m/2} \WhittakerM \left ( m/2,m/2+1/2,\mu \,y \right ) }{\mu \, \left ( m+1 \right ) }}}}{x}^{{\frac{m}{m+1}}} \left ({x}^{{\frac{mn}{m+1}}} \right ) ^{-1} \left ({x}^{{\frac{n}{m+1}}} \right ) ^{-1} \left ({{\rm e}^{{\frac{b{{\rm e}^{-1/2\,\mu \,y}}{y}^{m} \left ( \mu \,y \right ) ^{-m/2} \WhittakerM \left ( m/2,m/2+1/2,\mu \,y \right ) }{\mu \, \left ( m+1 \right ) }}}} \right ) ^{-1}}+an\int \!{{\rm e}^{{\frac{bn{{\rm e}^{-1/2\,\mu \,y}}{y}^{m} \left ( \mu \,y \right ) ^{-m/2} \WhittakerM \left ( m/2,m/2+1/2,\mu \,y \right ) -b{{\rm e}^{-1/2\,\mu \,y}}{y}^{m} \left ( \mu \,y \right ) ^{-m/2} \WhittakerM \left ( m/2,m/2+1/2,\mu \,y \right ) +\lambda \,\mu \,ym-{\mu }^{2}ym+\lambda \,\mu \,y-{\mu }^{2}y}{\mu \, \left ( m+1 \right ) }}}}\,{\rm d}y-a\int \!{{\rm e}^{{\frac{bn{{\rm e}^{-1/2\,\mu \,y}}{y}^{m} \left ( \mu \,y \right ) ^{-m/2} \WhittakerM \left ( m/2,m/2+1/2,\mu \,y \right ) -b{{\rm e}^{-1/2\,\mu \,y}}{y}^{m} \left ( \mu \,y \right ) ^{-m/2} \WhittakerM \left ( m/2,m/2+1/2,\mu \,y \right ) +\lambda \,\mu \,ym-{\mu }^{2}ym+\lambda \,\mu \,y-{\mu }^{2}y}{\mu \, \left ( m+1 \right ) }}}}\,{\rm d}y \right ) \]

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41.35 problem number 35

problem number 371

Added January 10, 2019.

Problem 2.3.2.35 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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41.36 problem number 36

problem number 372

Added January 10, 2019.

Problem 2.3.2.36 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({{x}^{ \left ( k+1 \right ) ^{-1}}{{\rm e}^{{\frac{bn{{\rm e}^{-1/2\,y\lambda }}{y}^{k} \left ( y\lambda \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,y\lambda \right ) }{\lambda \, \left ( k+1 \right ) }}}}{x}^{{\frac{k}{k+1}}} \left ({x}^{{\frac{kn}{k+1}}} \right ) ^{-1} \left ({x}^{{\frac{n}{k+1}}} \right ) ^{-1} \left ({{\rm e}^{{\frac{b{{\rm e}^{-1/2\,y\lambda }}{y}^{k} \left ( y\lambda \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,y\lambda \right ) }{\lambda \, \left ( k+1 \right ) }}}} \right ) ^{-1}}+an\int \!{{\rm e}^{{\frac{bn{{\rm e}^{-1/2\,y\lambda }}{y}^{k} \left ( y\lambda \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,y\lambda \right ) -b{{\rm e}^{-1/2\,y\lambda }}{y}^{k} \left ( y\lambda \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,y\lambda \right ) -{\lambda }^{2}yk-{\lambda }^{2}y}{\lambda \, \left ( k+1 \right ) }}}}{y}^{m}\,{\rm d}y-a\int \!{{\rm e}^{{\frac{bn{{\rm e}^{-1/2\,y\lambda }}{y}^{k} \left ( y\lambda \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,y\lambda \right ) -b{{\rm e}^{-1/2\,y\lambda }}{y}^{k} \left ( y\lambda \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,y\lambda \right ) -{\lambda }^{2}yk-{\lambda }^{2}y}{\lambda \, \left ( k+1 \right ) }}}}{y}^{m}\,{\rm d}y \right ) \]