72 HFOPDE, chapter 3.3.1

 72.1 Problem 1
 72.2 Problem 2
 72.3 Problem 3
 72.4 Problem 4
 72.5 Problem 5
 72.6 Problem 6
 72.7 Problem 7
 72.8 Problem 8
 72.9 Problem 9
 72.10 Problem 10
 72.11 Problem 11

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72.1 Problem 1

problem number 670

Added Feb. 9, 2019.

Problem Chapter 3.3.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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72.2 Problem 2

problem number 671

Added Feb. 9, 2019.

Problem Chapter 3.3.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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72.3 Problem 3

problem number 672

Added Feb. 9, 2019.

Problem Chapter 3.3.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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72.4 Problem 4

problem number 673

Added Feb. 9, 2019.

Problem Chapter 3.3.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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72.5 Problem 5

problem number 674

Added Feb. 9, 2019.

Problem Chapter 3.3.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a e^{\alpha x} w_x + b e^{\beta y} w_y = c e^{\gamma x-\beta y} \]

Mathematica

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

Maple

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

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72.6 Problem 6

problem number 675

Added Feb. 9, 2019.

Problem Chapter 3.3.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a e^{\alpha x} w_x + b e^{\beta y} w_y = c e^{\gamma x-2 \beta y} \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{e^{-2 \beta y} \left (6 a^3 \alpha ^3 e^{2 \beta y} c_1\left (-\frac{e^{-\alpha x-\beta y} \left (a \alpha e^{\alpha x}-b \beta e^{\beta y}\right )}{a \alpha \beta }\right )-11 a^3 \alpha ^2 \gamma e^{2 \beta y} c_1\left (-\frac{e^{-\alpha x-\beta y} \left (a \alpha e^{\alpha x}-b \beta e^{\beta y}\right )}{a \alpha \beta }\right )-a^3 \gamma ^3 e^{2 \beta y} c_1\left (-\frac{e^{-\alpha x-\beta y} \left (a \alpha e^{\alpha x}-b \beta e^{\beta y}\right )}{a \alpha \beta }\right )+6 a^3 \alpha \gamma ^2 e^{2 \beta y} c_1\left (-\frac{e^{-\alpha x-\beta y} \left (a \alpha e^{\alpha x}-b \beta e^{\beta y}\right )}{a \alpha \beta }\right )-6 a^2 \alpha ^2 c e^{x (\gamma -3 \alpha )+2 \alpha x}-a^2 c \gamma ^2 e^{x (\gamma -3 \alpha )+2 \alpha x}+5 a^2 \alpha c \gamma e^{x (\gamma -3 \alpha )+2 \alpha x}+6 a \alpha b \beta c e^{x (\gamma -3 \alpha )+\alpha x+\beta y}-2 a b \beta c \gamma e^{x (\gamma -3 \alpha )+\alpha x+\beta y}-2 b^2 \beta ^2 c e^{x (\gamma -3 \alpha )+2 \beta y}\right )}{a^3 (\alpha -\gamma ) (2 \alpha -\gamma ) (3 \alpha -\gamma )}\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\frac{{b}^{2}{\beta }^{2}c}{{a}^{3}{\alpha }^{2}} \left ({\frac{ \left ({{\rm e}^{\beta \,y}}b\beta -a\alpha \,{{\rm e}^{\alpha \,x}} \right ) ^{2}{{\rm e}^{-2\,\alpha \,x-2\,\beta \,y+x \left ( \gamma -\alpha \right ) }}}{{b}^{2}{\beta }^{2} \left ( \gamma -\alpha \right ) }}-2\,{\frac{ \left ({{\rm e}^{\beta \,y}}b\beta -a\alpha \,{{\rm e}^{\alpha \,x}} \right ){{\rm e}^{-\alpha \,x-\beta \,y+x \left ( \gamma -2\,\alpha \right ) }}}{b\beta \, \left ( \gamma -2\,\alpha \right ) }}+{\frac{{{\rm e}^{x \left ( \gamma -3\,\alpha \right ) }}}{\gamma -3\,\alpha }} \right ) }+{\it \_F1} \left ({\frac{ \left ({{\rm e}^{\beta \,y}}b\beta -a\alpha \,{{\rm e}^{\alpha \,x}} \right ){{\rm e}^{-\alpha \,x-\beta \,y}}}{\alpha \,b\beta }} \right ) \]

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72.7 Problem 7

problem number 676

Added Feb. 9, 2019.

Problem Chapter 3.3.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a e^{\alpha x} w_x + b e^{\beta y} w_y = c e^{\gamma x} + s e^{\mu y} \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\frac{c{{\rm e}^{x \left ( \gamma -\alpha \right ) }}}{a \left ( \gamma -\alpha \right ) }}+{\frac{s \left ({{\rm e}^{\beta \,y}}b\beta -a\alpha \,{{\rm e}^{\alpha \,x}} \right ){{\rm e}^{-\alpha \,x-\beta \,y}}}{a\alpha \,b \left ( \beta -\mu \right ) } \left ({\frac{a\alpha }{b\beta } \left ( -{\frac{ \left ({{\rm e}^{\beta \,y}}b\beta -a\alpha \,{{\rm e}^{\alpha \,x}} \right ){{\rm e}^{-\alpha \,x-\beta \,y}}}{b\beta }}+{{\rm e}^{-\alpha \,x}} \right ) ^{-1}} \right ) ^{{\frac{\mu }{\beta }}}}-{\frac{\beta \,s{{\rm e}^{-\alpha \,x}}}{a\alpha \, \left ( \beta -\mu \right ) } \left ({\frac{a\alpha }{b\beta } \left ( -{\frac{ \left ({{\rm e}^{\beta \,y}}b\beta -a\alpha \,{{\rm e}^{\alpha \,x}} \right ){{\rm e}^{-\alpha \,x-\beta \,y}}}{b\beta }}+{{\rm e}^{-\alpha \,x}} \right ) ^{-1}} \right ) ^{{\frac{\mu }{\beta }}}}+{\it \_F1} \left ({\frac{ \left ({{\rm e}^{\beta \,y}}b\beta -a\alpha \,{{\rm e}^{\alpha \,x}} \right ){{\rm e}^{-\alpha \,x-\beta \,y}}}{\alpha \,b\beta }} \right ) \]

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72.8 Problem 8

problem number 677

Added Feb. 9, 2019.

Problem Chapter 3.3.1.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \text{DSolve}\left [a e^{\beta x} w^{(1,0)}(x,y)+w^{(0,1)}(x,y) \left (b e^{\gamma x}+c e^{\lambda y}\right )=k e^{\delta y}+s e^{\mu x}+p,w(x,y),\{x,y\}\right ] \]

Maple

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

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72.9 Problem 9

problem number 678

Added Feb. 9, 2019.

Problem Chapter 3.3.1.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \text{DSolve}\left [a e^{\beta x} w^{(1,0)}(x,y)+w^{(0,1)}(x,y) \left (b e^{\gamma x}+c e^{\lambda y}\right )=s e^{\delta y+\mu x}+k,w(x,y),\{x,y\}\right ] \]

Maple

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

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72.10 Problem 10

problem number 679

Added Feb. 9, 2019.

Problem Chapter 3.3.1.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{e^{-\beta x} \left (-\beta c e^{x (\mu -\beta )+\beta x} \left (-\frac{a (\beta -\gamma ) e^{\beta x}}{-e^{-\lambda y} \left (-a \gamma e^{\beta x}+a \beta e^{\beta x}-b \lambda e^{\gamma x+\lambda y}\right )-b \lambda e^{\gamma x}}\right )^{\delta /\lambda } \left (\frac{b e^{x (\gamma -\beta )}}{\frac{\beta e^{-\beta x-\lambda y} \left (-a \gamma e^{\beta x}+a \beta e^{\beta x}-b \lambda e^{\gamma x+\lambda y}\right )}{\lambda (\beta -\gamma )}-\frac{\gamma e^{-\beta x-\lambda y} \left (-a \gamma e^{\beta x}+a \beta e^{\beta x}-b \lambda e^{\gamma x+\lambda y}\right )}{\lambda (\beta -\gamma )}}+1\right )^{\delta /\lambda } \text{Hypergeometric2F1}\left (\frac{\delta }{\lambda },\frac{\beta -\mu }{\beta -\gamma },\frac{\beta -\mu }{\beta -\gamma }+1,\frac{b e^{\gamma x-\beta x}}{\frac{\gamma e^{-\beta x-\lambda y} \left (-a \gamma e^{\beta x}+a \beta e^{\beta x}-b \lambda e^{\gamma x+\lambda y}\right )}{\lambda (\beta -\gamma )}-\frac{\beta e^{-\beta x-\lambda y} \left (-a \gamma e^{\beta x}+a \beta e^{\beta x}-b \lambda e^{\gamma x+\lambda y}\right )}{\lambda (\beta -\gamma )}}\right )+a \beta ^2 e^{\beta x} c_1\left (-\frac{e^{-\beta x-\lambda y} \left (-a \gamma e^{\beta x}+a \beta e^{\beta x}-b \lambda e^{\gamma x+\lambda y}\right )}{a \lambda (\beta -\gamma )}\right )-a \beta \mu e^{\beta x} c_1\left (-\frac{e^{-\beta x-\lambda y} \left (-a \gamma e^{\beta x}+a \beta e^{\beta x}-b \lambda e^{\gamma x+\lambda y}\right )}{a \lambda (\beta -\gamma )}\right )-\beta k+k \mu \right )}{a \beta (\beta -\mu )}\right \}\right \} \]

Maple

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

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72.11 Problem 11

problem number 680

Added Feb. 9, 2019.

Problem Chapter 3.3.1.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a e^{\lambda x} w_x + b e^{\beta x} w_y = c e^{\gamma x} + d \]

Mathematica

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

Maple

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