73 HFOPDE, chapter 3.3.2

 73.1 Problem 1
 73.2 Problem 2
 73.3 Problem 3
 73.4 Problem 4
 73.5 Problem 5
 73.6 Problem 6
 73.7 Problem 7
 73.8 Problem 8
 73.9 Problem 9
 73.10 Problem 10
 73.11 Problem 11

_______________________________________________________________________________________

73.1 Problem 1

problem number 681

Added Feb. 9, 2019.

Problem Chapter 3.3.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c y e^{\lambda x} + k x e^{\mu y} \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\frac{kx}{\mu \,b}{{\rm e}^{{\frac{\mu \, \left ( ay-bx \right ) }{a}}+{\frac{\mu \,bx}{a}}}}}+{\frac{cy{{\rm e}^{\lambda \,x}}}{\lambda \,a}}-{\frac{1}{{\mu }^{2}{b}^{2}{a}^{2}{\lambda }^{2}} \left ( -{\it \_F1} \left ({\frac{ay-bx}{a}} \right ){\mu }^{2}{b}^{2}{a}^{2}{\lambda }^{2}+{a}^{3}k{{\rm e}^{{\frac{\mu \, \left ( ay-bx \right ) }{a}}+{\frac{\mu \,bx}{a}}}}{\lambda }^{2}+{{\rm e}^{\lambda \,x}}{b}^{3}c{\mu }^{2} \right ) } \]

_______________________________________________________________________________________

73.2 Problem 2

problem number 682

Added Feb. 9, 2019.

Problem Chapter 3.3.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y = a x^k e^{\lambda y} \]

Mathematica

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

Maple

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

_______________________________________________________________________________________

73.3 Problem 3

problem number 683

Added Feb. 9, 2019.

Problem Chapter 3.3.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a y+b e^{\lambda x}) w_y = c e^{\beta x} \]

Mathematica

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

Maple

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

_______________________________________________________________________________________

73.4 Problem 4

problem number 684

Added Feb. 9, 2019.

Problem Chapter 3.3.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a y e^{\lambda x}+b e^{\beta x} y^k) w_y = c e^{\mu x} \]

Mathematica

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

Maple

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

_______________________________________________________________________________________

73.5 Problem 5

problem number 685

Added Feb. 9, 2019.

Problem Chapter 3.3.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\frac{1}{\beta } \left ( c{{\rm e}^{\beta \,x}}+{\it \_F1} \left ({\frac{1}{a\lambda \, \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{\lambda \, \left ({x}^{k+1}a-ky-y \right ) }{k+1}}}}a+5\,{x}^{-k+n} \left ( -{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac{{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( -1/2\,{\frac{k-n}{k+1}},1/2\,{\frac{n+3+2\,k}{k+1}},-{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) bk+4\,{x}^{-k+n} \left ( -{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac{{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( -1/2\,{\frac{k-n}{k+1}},1/2\,{\frac{n+3+2\,k}{k+1}},-{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) b{k}^{2}+{x}^{-k+n} \left ( -{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac{{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( -1/2\,{\frac{k-n}{k+1}},1/2\,{\frac{n+3+2\,k}{k+1}},-{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) b{k}^{3}+{x}^{-k+n} \left ( -{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac{{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( -1/2\,{\frac{k-n}{k+1}},1/2\,{\frac{n+3+2\,k}{k+1}},-{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) bn+8\,{x}^{-k+n} \left ( -{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac{{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( 1/2\,{\frac{k+n+2}{k+1}},1/2\,{\frac{n+3+2\,k}{k+1}},-{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) bk+5\,{x}^{-k+n} \left ( -{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac{{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( 1/2\,{\frac{k+n+2}{k+1}},1/2\,{\frac{n+3+2\,k}{k+1}},-{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) b{k}^{2}+{x}^{-k+n} \left ( -{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac{{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( 1/2\,{\frac{k+n+2}{k+1}},1/2\,{\frac{n+3+2\,k}{k+1}},-{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) b{k}^{3}+4\,{x}^{-k+n} \left ( -{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac{{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( 1/2\,{\frac{k+n+2}{k+1}},1/2\,{\frac{n+3+2\,k}{k+1}},-{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) bn+{x}^{-k+n} \left ( -{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac{{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( 1/2\,{\frac{k+n+2}{k+1}},1/2\,{\frac{n+3+2\,k}{k+1}},-{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) b{n}^{2}- \left ( -{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac{{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( -1/2\,{\frac{k-n}{k+1}},1/2\,{\frac{n+3+2\,k}{k+1}},-{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) ab{k}^{2}\lambda \,{x}^{n+1}-11\,{{\rm e}^{{\frac{\lambda \, \left ({x}^{k+1}a-ky-y \right ) }{k+1}}}}an-2\,{{\rm e}^{{\frac{\lambda \, \left ({x}^{k+1}a-ky-y \right ) }{k+1}}}}a{k}^{2}-6\,{{\rm e}^{{\frac{\lambda \, \left ({x}^{k+1}a-ky-y \right ) }{k+1}}}}a{n}^{2}-7\,{{\rm e}^{{\frac{\lambda \, \left ({x}^{k+1}a-ky-y \right ) }{k+1}}}}ak-{{\rm e}^{{\frac{\lambda \, \left ({x}^{k+1}a-ky-y \right ) }{k+1}}}}a{n}^{3}-10\,{{\rm e}^{{\frac{\lambda \, \left ({x}^{k+1}a-ky-y \right ) }{k+1}}}}akn-2\,{{\rm e}^{{\frac{\lambda \, \left ({x}^{k+1}a-ky-y \right ) }{k+1}}}}a{k}^{2}n-3\,{{\rm e}^{{\frac{\lambda \, \left ({x}^{k+1}a-ky-y \right ) }{k+1}}}}ak{n}^{2}-2\, \left ( -{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac{{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( -1/2\,{\frac{k-n}{k+1}},1/2\,{\frac{n+3+2\,k}{k+1}},-{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) abk\lambda \,{x}^{n+1}+4\,{x}^{-k+n} \left ( -{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac{{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( 1/2\,{\frac{k+n+2}{k+1}},1/2\,{\frac{n+3+2\,k}{k+1}},-{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) b+2\,{x}^{-k+n} \left ( -{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac{{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( 1/2\,{\frac{k+n+2}{k+1}},1/2\,{\frac{n+3+2\,k}{k+1}},-{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) b{k}^{2}n+2\,{x}^{-k+n} \left ( -{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac{{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( -1/2\,{\frac{k-n}{k+1}},1/2\,{\frac{n+3+2\,k}{k+1}},-{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) bkn- \left ( -{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac{{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( -1/2\,{\frac{k-n}{k+1}},1/2\,{\frac{n+3+2\,k}{k+1}},-{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) ab\lambda \,{x}^{n+1}+{x}^{-k+n} \left ( -{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac{{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( -1/2\,{\frac{k-n}{k+1}},1/2\,{\frac{n+3+2\,k}{k+1}},-{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) b{k}^{2}n+6\,{x}^{-k+n} \left ( -{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac{{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( 1/2\,{\frac{k+n+2}{k+1}},1/2\,{\frac{n+3+2\,k}{k+1}},-{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) bkn+{x}^{-k+n} \left ( -{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac{{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( 1/2\,{\frac{k+n+2}{k+1}},1/2\,{\frac{n+3+2\,k}{k+1}},-{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) bk{n}^{2}+2\,{x}^{-k+n} \left ( -{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac{k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac{{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( -1/2\,{\frac{k-n}{k+1}},1/2\,{\frac{n+3+2\,k}{k+1}},-{\frac{{x}^{k+1}\lambda \,a}{k+1}} \right ) b \right ) } \right ) \beta \right ) } \]

_______________________________________________________________________________________

73.6 Problem 6

problem number 686

Added Feb. 9, 2019.

Problem Chapter 3.3.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = a x e^{\lambda x+ \mu y} \]

Mathematica

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

Maple

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

_______________________________________________________________________________________

73.7 Problem 7

problem number 687

Added Feb. 9, 2019.

Problem Chapter 3.3.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

_______________________________________________________________________________________

73.8 Problem 8

problem number 688

Added Feb. 9, 2019.

Problem Chapter 3.3.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^k w_x + b e^{\lambda y} w_y = c x^n+s \]

Mathematica

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

Maple

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

_______________________________________________________________________________________

73.9 Problem 9

problem number 689

Added Feb. 9, 2019.

Problem Chapter 3.3.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y^k w_x + b e^{\lambda x} w_y = c e^{\mu x}+s \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{\left (\left (\frac{(k+1) \left (\frac{a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}}{k+1}+b e^{\lambda x}\right )}{a \lambda }\right )^{\frac{1}{k+1}}\right )^{-k} \left (c k \lambda e^{\mu x} \left (\frac{b (k+1) e^{\lambda x}}{a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}}+1\right )^{\frac{k}{k+1}} \text{Hypergeometric2F1}\left (\frac{k}{k+1},\frac{\mu }{\lambda },\frac{\lambda +\mu }{\lambda },-\frac{b (k+1) e^{\lambda x}}{a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}}\right )-k \mu s \left (\frac{e^{-\lambda x} \left (a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}\right )}{b (k+1)}+1\right )^{\frac{k}{k+1}} \text{Hypergeometric2F1}\left (\frac{k}{k+1},\frac{k}{k+1},\frac{k}{k+1}+1,-\frac{e^{-\lambda x} \left (a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}\right )}{b (k+1)}\right )-\mu s \left (\frac{e^{-\lambda x} \left (a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}\right )}{b (k+1)}+1\right )^{\frac{k}{k+1}} \text{Hypergeometric2F1}\left (\frac{k}{k+1},\frac{k}{k+1},\frac{k}{k+1}+1,-\frac{e^{-\lambda x} \left (a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}\right )}{b (k+1)}\right )+a k \lambda \mu \left (\left (\frac{(k+1) \left (\frac{a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}}{k+1}+b e^{\lambda x}\right )}{a \lambda }\right )^{\frac{1}{k+1}}\right )^k c_1\left (\frac{a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}}{a (k+1) \lambda }\right )\right )}{a k \lambda \mu }\right \}\right \} \]

Maple

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

_______________________________________________________________________________________

73.10 Problem 10

problem number 690

Added Feb. 9, 2019.

Problem Chapter 3.3.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a e^{\lambda x} w_x + b y^k w_y = c x^n+s \]

Mathematica

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

Maple

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

_______________________________________________________________________________________

73.11 Problem 11

problem number 691

Added Feb. 9, 2019.

Problem Chapter 3.3.2.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a e^{\lambda y} w_x + b x^k w_y = c Exp[\mu x]+s \]

Mathematica

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

Maple

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