6.5.5 3.1

6.5.5.1 [1231] Problem 1

problem number 1231

Added April 1, 2019.

Problem Chapter 5.3.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.5.2 [1232] Problem 2

problem number 1232

Added April 1, 2019.

Problem Chapter 5.3.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.5.3 [1233] Problem 3

problem number 1233

Added April 1, 2019.

Problem Chapter 5.3.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.5.4 [1234] Problem 4

problem number 1234

Added April 1, 2019.

Problem Chapter 5.3.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*w[x,y] + k*Exp[mu*x+delta*y]; 
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*w(x,y)+ k*exp(mu*x+delta*y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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6.5.5.5 [1235] Problem 5

problem number 1235

Added April 1, 2019.

Problem Chapter 5.3.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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]*w[x,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)*w(x,y)+k; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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6.5.5.6 [1236] Problem 6

problem number 1236

Added April 1, 2019.

Problem Chapter 5.3.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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[sigma*y]*w[x,y]+k*Exp[mu*x+delta*y]+d; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \exp \left (\frac {c (\gamma -\beta ) \left (e^{\lambda y}\right )^{\frac {\sigma }{\lambda }} e^{-\gamma x-\lambda y} \, _2F_1\left (1,-\frac {\gamma }{\beta -\gamma };\frac {\beta \sigma -\gamma (\lambda +\sigma )}{(\beta -\gamma ) \lambda };1-\frac {a e^{\beta x-\gamma x-\lambda y} (\beta -\gamma )}{b \lambda }\right )}{b (\beta (\lambda -\sigma )+\gamma \sigma )}\right ) \left (\int _1^x\frac {\exp \left (\frac {c e^{-\gamma K[1]} (\beta -\gamma ) \left (\frac {a e^{\lambda y+\beta (x+K[1])} (\beta -\gamma )}{a e^{\beta (x+K[1])} (\beta -\gamma )-b e^{\lambda y} \left (e^{\gamma x+\beta K[1]}-e^{\beta x+\gamma K[1]}\right ) \lambda }\right )^{\frac {\sigma }{\lambda }-1} \, _2F_1\left (1,-\frac {\gamma }{\beta -\gamma };\frac {\beta \sigma -\gamma (\lambda +\sigma )}{(\beta -\gamma ) \lambda };e^{(\beta -\gamma ) K[1]} \left (\frac {a e^{-\lambda y} (\gamma -\beta )}{b \lambda }+e^{(\gamma -\beta ) x}\right )\right )}{b (\beta (\lambda -\sigma )+\gamma \sigma )}-\beta K[1]\right ) \left (e^{\mu K[1]} k \left (\frac {a e^{\lambda y+\beta (x+K[1])} (\gamma -\beta )}{b e^{\lambda y} \left (e^{\gamma x+\beta K[1]}-e^{\beta x+\gamma K[1]}\right ) \lambda -a e^{\beta (x+K[1])} (\beta -\gamma )}\right )^{\delta /\lambda }+d\right )}{a}dK[1]+c_1\left (\frac {b e^{\gamma x-\beta x}}{a \beta -a \gamma }-\frac {e^{-\lambda y}}{\lambda }\right )\right )\right \}\right \}\]

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(sigma*y)*w(x,y)+k*exp(mu*x+delta*y)+d; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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6.5.5.7 [1237] Problem 7

problem number 1237

Added April 1, 2019.

Problem Chapter 5.3.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.5.8 [1238] Problem 8

problem number 1238

Added April 1, 2019.

Problem Chapter 5.3.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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