7.3.6 3.1

7.3.6.1 [879] Problem 1

problem number 879

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 ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == c*Exp[lambda*x] + d*Exp[mu*y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=a*diff(w(x,y),x) +b*diff(w(x,y),y) =c*exp(lambda*x)+d*exp(mu*y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = \frac {a b \lambda \mu \textit {\_F1} \left (\frac {a y -b x}{a}\right )+a d \lambda \,{\mathrm e}^{\mu y}+b c \mu \,{\mathrm e}^{\lambda x}}{a b \lambda \mu }\]

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7.3.6.2 [880] Problem 2

problem number 880

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 ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == c*Exp[lambda*x + beta*y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=a*diff(w(x,y),x) +b*diff(w(x,y),y) =c*exp(lambda*x+beta*y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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7.3.6.3 [881] Problem 3

problem number 881

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 ✓

ClearAll["Global`*"]; 
pde =  a*Exp[lambda*x]*D[w[x, y], x] + b*Exp[beta*y]*D[w[x, y], y] == c; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=a*exp(lambda*x)*diff(w(x,y),x) +b*exp(beta*y)*diff(w(x,y),y) =c; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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7.3.6.4 [882] Problem 4

problem number 882

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 ✓

ClearAll["Global`*"]; 
pde =  a*Exp[lambda*y]*D[w[x, y], x] + b*Exp[beta*x]*D[w[x, y], y] == c; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=a*exp(lambda*y)*diff(w(x,y),x) +b*exp(beta*x)*diff(w(x,y),y) =c; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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7.3.6.5 [883] Problem 5

problem number 883

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 ✓

ClearAll["Global`*"]; 
pde =  a*Exp[alpha*x]*D[w[x, y], x] + b*Exp[beta*y]*D[w[x, y], y] == c*Exp[gamma*x - beta*y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=a*exp(alpha*x)*diff(w(x,y),x) +b*exp(beta*y)*diff(w(x,y),y) =c*exp(gamma*x-beta*y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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7.3.6.6 [884] Problem 6

problem number 884

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 ✓

ClearAll["Global`*"]; 
pde =  a*Exp[alpha*x]*D[w[x, y], x] + b*Exp[beta*y]*D[w[x, y], y] == c*Exp[gamma*x - 2*beta*y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=a*exp(alpha*x)*diff(w(x,y),x) +b*exp(beta*y)*diff(w(x,y),y) =c*exp(gamma*x-2*beta*y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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7.3.6.7 [885] Problem 7

problem number 885

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 ✓

ClearAll["Global`*"]; 
pde =  a*Exp[alpha*x]*D[w[x, y], x] + b*Exp[beta*y]*D[w[x, y], y] == c*Exp[gamma*x] + s*Exp[mu*y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {b e^{-\alpha x}}{a \alpha }-\frac {e^{-\beta y}}{\beta }\right )-\frac {c e^{\gamma x-\alpha x}}{a \alpha -a \gamma }-\frac {s \left (e^{-\beta y}\right )^{1-\frac {\mu }{\beta }}}{b \beta -b \mu }\right \}\right \}\]

Maple ✓

restart; 
pde :=a*exp(alpha*x)*diff(w(x,y),x) +b*exp(beta*y)*diff(w(x,y),y) =c*exp(gamma*x) + s*exp(mu*y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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7.3.6.8 [886] Problem 8

problem number 886

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 ✗

ClearAll["Global`*"]; 
pde =  a*Exp[beta*x]*D[w[x, y], x] + (b*Exp[gamma*x] + c*Exp[lambda*y])*D[w[x, y], y] == s*Exp[mu*x] + k*Exp[delta*y] + p; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=a*exp(beta*x)*diff(w(x,y),x) +(b*exp(gamma*x)+c*exp(lambda*y))*diff(w(x,y),y) =s*exp(mu*x) + k*exp(delta*y)+p; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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7.3.6.9 [887] Problem 9

problem number 887

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 ✗

ClearAll["Global`*"]; 
pde =  a*Exp[beta*x]*D[w[x, y], x] + (b*Exp[gamma*x] + c*Exp[lambda*y])*D[w[x, y], y] == s*Exp[mu*x + delta*y] + k; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=a*exp(beta*x)*diff(w(x,y),x) +(b*exp(gamma*x)+c*exp(lambda*y))*diff(w(x,y),y) =s*exp(mu*x+delta*y)+k; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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7.3.6.10 [888] Problem 10

problem number 888

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 ✓

ClearAll["Global`*"]; 
pde =  a*Exp[beta*x]*D[w[x, y], x] + b*Exp[gamma*x + lambda*y]*D[w[x, y], y] == c*Exp[mu*x + delta*y] + k; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {b e^{\gamma x-\beta x}}{a \beta -a \gamma }-\frac {e^{-\lambda y}}{\lambda }\right )+\frac {c (\gamma -\beta ) \left (e^{\lambda y}\right )^{\delta /\lambda } e^{-\gamma x-\lambda y+\mu x} \, _2F_1\left (1,\frac {\mu -\gamma }{\beta -\gamma };\frac {\beta \delta -\gamma \delta -\gamma \lambda +\lambda \mu }{\beta \lambda -\gamma \lambda };1-\frac {a e^{\beta x-\gamma x-\lambda y} (\beta -\gamma )}{b \lambda }\right )}{b (\beta (\lambda -\delta )+\delta \gamma -\lambda \mu )}-\frac {k e^{-\beta x}}{a \beta }\right \}\right \}\]

Maple ✓

restart; 
pde :=a*exp(beta*x)*diff(w(x,y),x) +b*exp(gamma*x+lambda*y)*diff(w(x,y),y) =c*exp(mu*x+delta*y)+k; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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7.3.6.11 [889] Problem 11

problem number 889

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 ✓

ClearAll["Global`*"]; 
pde =  a*Exp[lambda*y]*D[w[x, y], x] + b*Exp[beta*x]*D[w[x, y], y] == c*Exp[gamma*y] + d; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{\lambda y}}{\lambda }-\frac {b e^{\beta x}}{a \beta }\right )-\frac {d \gamma \log \left (\frac {a \beta e^{\lambda y}}{\lambda }\right )-\beta d \gamma x+c \lambda \left (e^{\lambda y}\right )^{\gamma /\lambda }}{a \beta \gamma e^{\lambda y}-b \gamma \lambda e^{\beta x}}-\frac {a \beta c \lambda \left (e^{\lambda y}\right )^{\frac {\gamma +\lambda }{\lambda }} \, _2F_1\left (1,\frac {\gamma +\lambda }{\lambda };\frac {\gamma }{\lambda }+2;\frac {a \beta e^{\lambda y}}{a \beta e^{\lambda y}-b e^{\beta x} \lambda }\right )}{(\gamma +\lambda ) \left (a \beta e^{\lambda y}-b \lambda e^{\beta x}\right )^2}\right \}\right \}\]

Maple ✓

restart; 
pde :=a*exp(lambda*y)*diff(w(x,y),x) +b*exp(beta*x)*diff(w(x,y),y) =c*exp(gamma*y)+d; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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