6.4.5 2.4

   6.4.5.1 [1045] Problem 1
   6.4.5.2 [1046] Problem 2 case \(n\o:not: =-1, n\o:not: =-2\)
   6.4.5.3 [1047] Problem 2 case \(n = -1\)
   6.4.5.4 [1048] Problem 2 case \(n = -2\)
   6.4.5.5 [1049] Problem 3
   6.4.5.6 [1050] Problem 4
   6.4.5.7 [1051] Problem 5
   6.4.5.8 [1052] Problem 6
   6.4.5.9 [1053] Problem 7
   6.4.5.10 [1054] Problem 8
   6.4.5.11 [1055] Problem 9
   6.4.5.12 [1056] Problem 10
   6.4.5.13 [1057] Problem 11
   6.4.5.14 [1058] Problem 12
   6.4.5.15 [1059] Problem 13

6.4.5.1 [1045] Problem 1

problem number 1045

Added Feb. 17, 2019.

Problem Chapter 4.2.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = (c x^n + d y^m) w \]

Mathematica

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

Maple

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

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6.4.5.2 [1046] Problem 2 case \(n\neq -1, n\neq -2\)

problem number 1046

Added Feb. 17, 2019.

Problem Chapter 4.2.4.2 case \(n eq -1, n eq -2\), from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = c x^n y w \]

Mathematica

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

Maple

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

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6.4.5.3 [1047] Problem 2 case \(n = -1\)

problem number 1047

Added Feb. 17, 2019.

Problem Chapter 4.2.4.2 case \(n= -1\), from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = c x^n y w \]

Mathematica

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

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ya-bx}{a}} \right ){x}^{{\frac{ \left ( ya-bx \right ) c}{{a}^{2}}}}{{\rm e}^{{\frac{cxb}{{a}^{2}}}}}\]

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6.4.5.4 [1048] Problem 2 case \(n = -2\)

problem number 1048

Added Feb. 17, 2019.

Problem Chapter 4.2.4.2 case \(n= -2\), from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = c x^n y w \]

Mathematica

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

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ya-bx}{a}} \right ){{\rm e}^{-{\frac{ \left ( ya-bx \right ) c}{{a}^{2}x}}}}{x}^{{\frac{bc}{{a}^{2}}}}\]

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6.4.5.5 [1049] Problem 3

problem number 1049

Added Feb. 17, 2019.

Problem Chapter 4.2.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x w_x + y w_y = a(x^2+y^2)^k w \]

Mathematica

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

Maple

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

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6.4.5.6 [1050] Problem 4

problem number 1050

Added Feb. 17, 2019.

Problem Chapter 4.2.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a x w_x + b y w_y = c x^n y^m w \]

Mathematica

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

Maple

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

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6.4.5.7 [1051] Problem 5

problem number 1051

Added Feb. 17, 2019.

Problem Chapter 4.2.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a x w_x + b y w_y = (c x^n + k y^m) w \]

Mathematica

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

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ( y{x}^{-{\frac{b}{a}}} \right ){{\rm e}^{{\frac{c{x}^{n}bm+k{y}^{m}an}{abmn}}}}\]

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6.4.5.8 [1052] Problem 6

problem number 1052

Added Feb. 17, 2019.

Problem Chapter 4.2.4.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ m x w_x + n y w_y = (a x^n + b y^m)^k w \]

Mathematica

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

Maple

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

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6.4.5.9 [1053] Problem 7

problem number 1053

Added Feb. 17, 2019.

Problem Chapter 4.2.4.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a x^n w_x + b y^m w_y = (c x^k + d y^s) w \]

Mathematica

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

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{-{x}^{-n+1}b \left ( m-1 \right ) +a \left ( n-1 \right ){y}^{-m+1}}{a \left ( n-1 \right ) }} \right ){{\rm e}^{{\frac{1}{ab \left ( k-n+1 \right ) \left ( m-s-1 \right ) } \left ( -{a}^{2}{y}^{-m+1}{a}^{{\frac{s}{m-1}}-1} \left ( n-1 \right ) ^{{\frac{s}{m-1}}}{{\rm e}^{{\frac{-i\pi \,s}{2\,m-2} \left ({\it csgn} \left ( i{y}^{-m+1}a \right ) \left ({\it csgn} \left ( i{y}^{-m+1}a \right ) -{\it csgn} \left ({\frac{i}{n-1}} \right ) \right ){\it csgn} \left ( i \left ( n-1 \right ){y}^{-m+1}a \right ) - \left ({\it csgn} \left ( i{y}^{-m+1}a \right ) \right ) ^{3}+{\it csgn} \left ({\frac{i}{n-1}} \right ) \left ({\it csgn} \left ( i{y}^{-m+1}a \right ) \right ) ^{2}+{\it csgn} \left ( i{y}^{-m+1} \right ) \left ({\it csgn} \left ( i{y}^{-m+1} \right ) -{\it csgn} \left ({\frac{i}{a}} \right ) \right ){\it csgn} \left ( i{y}^{-m+1}a \right ) - \left ({\it csgn} \left ( i{y}^{-m+1} \right ) \right ) ^{2} \left ({\it csgn} \left ( i{y}^{-m+1} \right ) -{\it csgn} \left ({\frac{i}{a}} \right ) \right ) \right ) }}}d \left ( k-n+1 \right ) \left ( a \left ( n-1 \right ){y}^{-m+1} \right ) ^{-{\frac{s}{m-1}}}+{x}^{k-n+1}cb \left ( m-s-1 \right ) \right ) }}}\]

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6.4.5.10 [1054] Problem 8

problem number 1054

Added Feb. 17, 2019.

Problem Chapter 4.2.4.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a x^n w_x + b x^m y w_y = (c x^k y^s + d) w \]

Mathematica

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

Maple

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

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6.4.5.11 [1055] Problem 9

problem number 1055

Added Feb. 17, 2019.

Problem Chapter 4.2.4.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a x^n w_x + (b x^m y+c x^k) w_y = (s x^p y^q + d) w \]

Mathematica

\[\left \{\left \{w(x,y)\to c_1\left (c (a (m-n+1))^{\frac{k-m}{m-n+1}} b^{\frac{-k+n-1}{m-n+1}} \text{Gamma}\left (\frac{k-n+1}{m-n+1},\frac{b x^{m-n+1}}{a m-a n+a}\right )+y e^{-\frac{b x^{m-n+1}}{a m-a n+a}}\right ) \exp \left (\int _1^x\frac{K[1]^{-n} \left (s \left (b^{-\frac{k+1}{m-n+1}} \exp \left (-\frac{b \left (x^{m-n+1}-K[1]^{m-n+1}\right )}{a (m-n+1)}\right ) (a (m-n+1))^{-\frac{m}{m-n+1}} \left (b^{\frac{n}{m-n+1}} c e^{\frac{b x^{m-n+1}}{m a-n a+a}} \text{Gamma}\left (\frac{k-n+1}{m-n+1},\frac{b x^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac{k}{m-n+1}}-b^{\frac{n}{m-n+1}} c e^{\frac{b x^{m-n+1}}{m a-n a+a}} \text{Gamma}\left (\frac{k-n+1}{m-n+1},\frac{b K[1]^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac{k}{m-n+1}}+b^{\frac{k+1}{m-n+1}} y (a (m-n+1))^{\frac{m}{m-n+1}}\right )\right )^q K[1]^p+d\right )}{a}dK[1]\right )\right \}\right \}\]

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{ab \left ( k+2\,m-3\,n+3 \right ) \left ( k+m-2\,n+2 \right ) \left ( k-n+1 \right ) } \left ( -a{{\rm e}^{-1/2\,{\frac{{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \left ({\frac{{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) ^{{\frac{-k-m+2\,n-2}{-2\,n+2\,m+2}}}c{x}^{k-m} \left ( -n+m+1 \right ) \left ( k+m-2\,n+2 \right ) ^{2} \WhittakerM \left ({\frac{k+m-2\,n+2}{-2\,n+2\,m+2}},{\frac{k+2\,m-3\,n+3}{-2\,n+2\,m+2}},{\frac{{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) - \left ({x}^{k-n+1}b+a{x}^{k-m} \left ( k+m-2\,n+2 \right ) \right ) \left ( -n+m+1 \right ) ^{2} \left ({\frac{{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) ^{{\frac{-k-m+2\,n-2}{-2\,n+2\,m+2}}}c{{\rm e}^{-1/2\,{\frac{{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \WhittakerM \left ({\frac{k-m}{-2\,n+2\,m+2}},{\frac{k+2\,m-3\,n+3}{-2\,n+2\,m+2}},{\frac{{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) +{{\rm e}^{-{\frac{{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}}yab \left ( k-n+1 \right ) \left ( k+2\,m-3\,n+3 \right ) \left ( k+m-2\,n+2 \right ) \right ) } \right ){{\rm e}^{\int ^{x}\!{\frac{1}{a} \left ({{\it \_a}}^{-n+p} \left ({\frac{1}{ab \left ( k+2\,m-3\,n+3 \right ) \left ( k+m-2\,n+2 \right ) \left ( k-n+1 \right ) } \left ( a{{\it \_a}}^{k-m}{{\rm e}^{1/2\,{\frac{{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \left ({\frac{{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) ^{{\frac{-k-m+2\,n-2}{-2\,n+2\,m+2}}}c \left ( -n+m+1 \right ) \left ( k+m-2\,n+2 \right ) ^{2} \WhittakerM \left ({\frac{k+m-2\,n+2}{-2\,n+2\,m+2}},{\frac{k+2\,m-3\,n+3}{-2\,n+2\,m+2}},{\frac{{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) -a{x}^{k-m}{{\rm e}^{{\frac{{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}}{{\rm e}^{-1/2\,{\frac{{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \left ({\frac{{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) ^{{\frac{-k-m+2\,n-2}{-2\,n+2\,m+2}}}c \left ( -n+m+1 \right ) \left ( k+m-2\,n+2 \right ) ^{2} \WhittakerM \left ({\frac{k+m-2\,n+2}{-2\,n+2\,m+2}},{\frac{k+2\,m-3\,n+3}{-2\,n+2\,m+2}},{\frac{{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) + \left ( -n+m+1 \right ) ^{2} \left ( b{{\it \_a}}^{k-n+1}+a{{\it \_a}}^{k-m} \left ( k+m-2\,n+2 \right ) \right ) \left ({\frac{{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) ^{{\frac{-k-m+2\,n-2}{-2\,n+2\,m+2}}}c{{\rm e}^{1/2\,{\frac{{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \WhittakerM \left ({\frac{k-m}{-2\,n+2\,m+2}},{\frac{k+2\,m-3\,n+3}{-2\,n+2\,m+2}},{\frac{{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) - \left ( \left ({x}^{k-n+1}b+a{x}^{k-m} \left ( k+m-2\,n+2 \right ) \right ) \left ( -n+m+1 \right ) ^{2} \left ({\frac{{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) ^{{\frac{-k-m+2\,n-2}{-2\,n+2\,m+2}}}c{{\rm e}^{-1/2\,{\frac{{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \WhittakerM \left ({\frac{k-m}{-2\,n+2\,m+2}},{\frac{k+2\,m-3\,n+3}{-2\,n+2\,m+2}},{\frac{{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) -{{\rm e}^{-{\frac{{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}}yab \left ( k-n+1 \right ) \left ( k+2\,m-3\,n+3 \right ) \left ( k+m-2\,n+2 \right ) \right ){{\rm e}^{{\frac{{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \right ) } \right ) ^{q}s+{{\it \_a}}^{-n}d \right ) }{d{\it \_a}}}}\]

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6.4.5.12 [1056] Problem 10

problem number 1056

Added Feb. 17, 2019.

Problem Chapter 4.2.4.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a x^n w_x + b x^m y^k w_y = (c x^p y^q + s) w \]

Mathematica

\[\left \{\left \{w(x,y)\to c_1\left (\frac{\frac{y^{1-k} (-m+n-1)}{k-1}-\frac{b x^{m-n+1}}{a}}{m-n+1}\right ) \exp \left (\frac{x^{1-n} \left (-\frac{c x^p \left (\left (y^{k-1}\right )^{\frac{1}{k-1}}\right )^q \left (\frac{a y (m-n+1) x^n}{a y (m-n+1) x^n+b (k-1) y^k x^{m+1}}\right )^{\frac{q}{k-1}} \, _2F_1\left (\frac{-n+p+1}{m-n+1},\frac{q}{k-1};\frac{m-2 n+p+2}{m-n+1};\frac{b (k-1) x^{m+1} y^k}{b (k-1) x^{m+1} y^k+a (m-n+1) x^n y}\right )}{n-p-1}-\frac{s}{n-1}\right )}{a}\right )\right \}\right \}\]

Maple

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

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6.4.5.13 [1057] Problem 11

problem number 1057

Added Feb. 17, 2019.

Problem Chapter 4.2.4.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a y^k w_x + b x^n w_y = (c x^m + s) w \]

Mathematica

\[\left \{\left \{w(x,y)\to c_1\left (\frac{y^{k+1}}{k+1}-\frac{b x^{n+1}}{a n+a}\right ) \exp \left (\frac{x \left (\left (y^{-k-1}\right )^{-\frac{1}{k+1}}\right )^{-k} \left (\frac{a (n+1) y^{k+1}}{a (n+1) y^{k+1}-b (k+1) x^{n+1}}\right )^{\frac{k}{k+1}} \left (c x^m \, _2F_1\left (\frac{k}{k+1},\frac{m+1}{n+1};\frac{m+n+2}{n+1};\frac{b (k+1) x^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )+(m+1) s \, _2F_1\left (\frac{k}{k+1},\frac{1}{n+1};1+\frac{1}{n+1};\frac{b (k+1) x^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )\right )}{a (m+1)}\right )\right \}\right \}\]

Maple

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

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6.4.5.14 [1058] Problem 12

problem number 1058

Added Feb. 17, 2019.

Problem Chapter 4.2.4.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x(x^n + (a n -1) y^n) w_x + y(y^n + (a n -1) x^n) w_y = k n (x^n + y^n) w \]

Mathematica

Failed

Maple

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

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6.4.5.15 [1059] Problem 13

problem number 1059

Added Feb. 17, 2019.

Problem Chapter 4.2.4.13, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x((n-2) y^n -2 x^n ) w_x + y(2 y^n - (n-2) x^n) w_y =\left ( (a(n-2)+2 b) y^n - (2 a + b(n-2)) x^n) \right ) w \]

Mathematica

Failed

Maple

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

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