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'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {k}{a}{{\rm e}^{{\frac {1}{a\lambda \,\mu \,b} \left ( -sa\lambda \,{{\rm e}^{{\frac {\mu \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}}}}+\mu \,b \left ( a\lambda \,{\it \_a}\,\nu -c{{\rm e}^{{\it \_a}\,\lambda }} \right ) \right ) }}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {c{{\rm e}^{x\lambda }}\mu \,b+sa\lambda \,{{\rm e}^{\mu \,y}}}{a\lambda \,\mu \,b}}}}\]
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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'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {k}{a}{{\rm e}^{{\frac {1}{a\alpha +\beta \,b} \left ( -c{{\rm e}^{{\frac { \left ( \alpha \,{\it \_a}+\beta \,y \right ) a-b\beta \, \left ( x-{\it \_a} \right ) }{a}}}}+{\it \_a}\,\gamma \, \left ( a\alpha +\beta \,b \right ) \right ) }}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {c{{\rm e}^{\alpha \,x+\beta \,y}}}{a\alpha +\beta \,b}}}}\]
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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'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {s}{a}{{\rm e}^{{\frac {1}{a \left ( -\beta +\lambda \right ) } \left ( -c \left ( -\beta +\lambda \right ) \int \!{{\rm e}^{{\frac {-\gamma \,{{\rm e}^{x \left ( \beta -\lambda \right ) }}b+\gamma \,{{\rm e}^{{\it \_b}\, \left ( \beta -\lambda \right ) }}b+a \left ( \beta -\lambda \right ) \left ( -{\it \_b}\,\lambda +\gamma \,y \right ) }{ \left ( \beta -\lambda \right ) a}}}}\,{\rm d}{\it \_b}-{{\rm e}^{{\it \_b}\, \left ( \beta -\lambda \right ) }}b\delta +{{\rm e}^{x \left ( \beta -\lambda \right ) }}b\delta -a \left ( -\beta +\lambda \right ) \left ( {\it \_b}\,\lambda -\mu \,{\it \_b}-\delta \,y \right ) \right ) }}}}{d{\it \_b}}+{\it \_F1} \left ( {\frac {-b{{\rm e}^{x \left ( \beta -\lambda \right ) }}+ay \left ( \beta -\lambda \right ) }{ \left ( \beta -\lambda \right ) a}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {c}{a}{{\rm e}^{{\frac {-\gamma \,{{\rm e}^{x \left ( \beta -\lambda \right ) }}b+\gamma \,{{\rm e}^{{\it \_a}\, \left ( \beta -\lambda \right ) }}b+a \left ( \beta -\lambda \right ) \left ( -{\it \_a}\,\lambda +\gamma \,y \right ) }{ \left ( \beta -\lambda \right ) a}}}}}{d{\it \_a}}}}\]
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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'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {k}{a} \left ( {\frac {1}{a} \left ( a\int \!{\frac {c}{a}{{\rm e}^{{\frac {-\lambda \,b{{\rm e}^{x \left ( \gamma -\beta \right ) }}-ax\beta \, \left ( -\gamma +\beta \right ) }{a \left ( -\gamma +\beta \right ) }}}}}\,{\rm d}x\lambda -c\int \!{{\rm e}^{{\frac {-\lambda \,b{{\rm e}^{{\it \_b}\, \left ( \gamma -\beta \right ) }}-a{\it \_b}\,\beta \, \left ( -\gamma +\beta \right ) }{a \left ( -\gamma +\beta \right ) }}}}\,{\rm d}{\it \_b}\lambda +a{{\rm e}^{-{\frac { \left ( b{{\rm e}^{x \left ( \gamma -\beta \right ) }}+ay \left ( -\gamma +\beta \right ) \right ) \lambda }{a \left ( -\gamma +\beta \right ) }}}} \right ) } \right ) ^{-{\frac {\delta }{\lambda }}}{{\rm e}^{{\frac {-{{\rm e}^{{\it \_b}\, \left ( \gamma -\beta \right ) }}b\beta \,\delta + \left ( -\gamma +\beta \right ) \left ( s{{\rm e}^{-\beta \,{\it \_b}}}+a{\it \_b}\,\beta \, \left ( -\beta +\mu \right ) \right ) }{a\beta \, \left ( -\gamma +\beta \right ) }}}}}{d{\it \_b}}+{\it \_F1} \left ( {\frac {1}{\lambda } \left ( -\lambda \,\int \!{\frac {c}{a}{{\rm e}^{{\frac {-\lambda \,b{{\rm e}^{x \left ( \gamma -\beta \right ) }}-ax\beta \, \left ( -\gamma +\beta \right ) }{a \left ( -\gamma +\beta \right ) }}}}}\,{\rm d}x-{{\rm e}^{-{\frac { \left ( b{{\rm e}^{x \left ( \gamma -\beta \right ) }}+ay \left ( -\gamma +\beta \right ) \right ) \lambda }{a \left ( -\gamma +\beta \right ) }}}} \right ) } \right ) \right ) {{\rm e}^{-{\frac {s{{\rm e}^{-\beta \,x}}}{a\beta }}}}\]
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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}{a}{{\rm e}^{{\frac {1}{a} \left ( -a\beta \,{\it \_f}-s\int \! \left ( {\frac {1}{a} \left ( a\int \!{\frac {c}{a}{{\rm e}^{{\frac {-\lambda \,b{{\rm e}^{x \left ( \gamma -\beta \right ) }}-ax\beta \, \left ( -\gamma +\beta \right ) }{a \left ( -\gamma +\beta \right ) }}}}}\,{\rm d}x\lambda -c\int \!{{\rm e}^{{\frac {-\lambda \,b{{\rm e}^{{\it \_f}\, \left ( \gamma -\beta \right ) }}-a{\it \_f}\,\beta \, \left ( -\gamma +\beta \right ) }{a \left ( -\gamma +\beta \right ) }}}}\,{\rm d}{\it \_f}\lambda +a{{\rm e}^{-{\frac { \left ( b{{\rm e}^{x \left ( \gamma -\beta \right ) }}+ay \left ( -\gamma +\beta \right ) \right ) \lambda }{a \left ( -\gamma +\beta \right ) }}}} \right ) } \right ) ^{-{\frac {\delta }{\lambda }}}{{\rm e}^{{\frac {-\delta \,b{{\rm e}^{{\it \_f}\, \left ( \gamma -\beta \right ) }}+a{\it \_f}\, \left ( -\gamma +\beta \right ) \left ( -\beta +\mu \right ) }{a \left ( -\gamma +\beta \right ) }}}}\,{\rm d}{\it \_f} \right ) }}}}{d{\it \_f}}+{\it \_F1} \left ( {\frac {1}{\lambda } \left ( -\lambda \,\int \!{\frac {c}{a}{{\rm e}^{{\frac {-\lambda \,b{{\rm e}^{x \left ( \gamma -\beta \right ) }}-ax\beta \, \left ( -\gamma +\beta \right ) }{a \left ( -\gamma +\beta \right ) }}}}}\,{\rm d}x-{{\rm e}^{-{\frac { \left ( b{{\rm e}^{x \left ( \gamma -\beta \right ) }}+ay \left ( -\gamma +\beta \right ) \right ) \lambda }{a \left ( -\gamma +\beta \right ) }}}} \right ) } \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {s}{a} \left ( {\frac {1}{a} \left ( a\int \!{\frac {c}{a}{{\rm e}^{{\frac {-\lambda \,b{{\rm e}^{x \left ( \gamma -\beta \right ) }}-ax\beta \, \left ( -\gamma +\beta \right ) }{a \left ( -\gamma +\beta \right ) }}}}}\,{\rm d}x\lambda -c\int \!{{\rm e}^{{\frac {-\lambda \,b{{\rm e}^{{\it \_b}\, \left ( \gamma -\beta \right ) }}-a{\it \_b}\,\beta \, \left ( -\gamma +\beta \right ) }{a \left ( -\gamma +\beta \right ) }}}}\,{\rm d}{\it \_b}\lambda +a{{\rm e}^{-{\frac { \left ( b{{\rm e}^{x \left ( \gamma -\beta \right ) }}+ay \left ( -\gamma +\beta \right ) \right ) \lambda }{a \left ( -\gamma +\beta \right ) }}}} \right ) } \right ) ^{-{\frac {\delta }{\lambda }}}{{\rm e}^{{\frac {-\delta \,b{{\rm e}^{{\it \_b}\, \left ( \gamma -\beta \right ) }}+a{\it \_b}\, \left ( -\gamma +\beta \right ) \left ( -\beta +\mu \right ) }{a \left ( -\gamma +\beta \right ) }}}}}{d{\it \_b}}}}\]
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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'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a} \left ( k \left ( {\frac {a \left ( -\gamma +\beta \right ) }{-b\lambda \,{{\rm e}^{-\lambda \,y}}{{\rm e}^{x \left ( \gamma -\beta \right ) +\lambda \,y}}+\lambda \,b{{\rm e}^{{\it \_b}\, \left ( \gamma -\beta \right ) }}+{{\rm e}^{-\lambda \,y}}a \left ( -\gamma +\beta \right ) }} \right ) ^{{\frac {\delta }{\lambda }}}{{\rm e}^{{\frac {1}{a} \left ( -c\int \! \left ( {\frac {a \left ( -\gamma +\beta \right ) }{-b\lambda \,{{\rm e}^{-\lambda \,y}}{{\rm e}^{x \left ( \gamma -\beta \right ) +\lambda \,y}}+\lambda \,b{{\rm e}^{{\it \_b}\, \left ( \gamma -\beta \right ) }}+{{\rm e}^{-\lambda \,y}}a \left ( -\gamma +\beta \right ) }} \right ) ^{{\frac {\sigma }{\lambda }}}{{\rm e}^{-\beta \,{\it \_b}}}\,{\rm d}{\it \_b}+a{\it \_b}\, \left ( -\beta +\mu \right ) \right ) }}}+d{{\rm e}^{{\frac {1}{a} \left ( -a{\it \_b}\,\beta -c\int \! \left ( {\frac {a \left ( -\gamma +\beta \right ) }{-b\lambda \,{{\rm e}^{-\lambda \,y}}{{\rm e}^{x \left ( \gamma -\beta \right ) +\lambda \,y}}+\lambda \,b{{\rm e}^{{\it \_b}\, \left ( \gamma -\beta \right ) }}+{{\rm e}^{-\lambda \,y}}a \left ( -\gamma +\beta \right ) }} \right ) ^{{\frac {\sigma }{\lambda }}}{{\rm e}^{-\beta \,{\it \_b}}}\,{\rm d}{\it \_b} \right ) }}} \right ) }{d{\it \_b}}+{\it \_F1} \left ( -{\frac {{{\rm e}^{-\lambda \,y}} \left ( -b\lambda \,{{\rm e}^{x \left ( \gamma -\beta \right ) +\lambda \,y}}+a \left ( -\gamma +\beta \right ) \right ) }{b\lambda \, \left ( -\gamma +\beta \right ) }} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {c{{\rm e}^{-\beta \,{\it \_a}}}}{a} \left ( {\frac {a \left ( -\gamma +\beta \right ) }{-b\lambda \,{{\rm e}^{-\lambda \,y}}{{\rm e}^{x \left ( \gamma -\beta \right ) +\lambda \,y}}+\lambda \,b{{\rm e}^{{\it \_a}\, \left ( \gamma -\beta \right ) }}+{{\rm e}^{-\lambda \,y}}a \left ( -\gamma +\beta \right ) }} \right ) ^{{\frac {\sigma }{\lambda }}}}{d{\it \_a}}}}\]
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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'));
\[w \left ( x,y \right ) = \left ( \int \!{\frac {s}{a}{{\rm e}^{{\frac {c{{\rm e}^{-x\lambda }}+ax\lambda \, \left ( \gamma -\lambda \right ) }{a\lambda }}}}}\,{\rm d}x+{\it \_F1} \left ( {\frac {-b{{\rm e}^{x \left ( \beta -\lambda \right ) }}+ay \left ( \beta -\lambda \right ) }{ \left ( \beta -\lambda \right ) a}} \right ) \right ) {{\rm e}^{-{\frac {c{{\rm e}^{-x\lambda }}}{a\lambda }}}}\]
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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'));
\[w \left ( x,y \right ) = \left ( \int \!{\frac {s}{a}{{\rm e}^{{\frac {-c{{\rm e}^{ \left ( \gamma -\lambda \right ) x}}-ax\lambda \, \left ( \gamma -\lambda \right ) }{ \left ( \gamma -\lambda \right ) a}}}}}\,{\rm d}x+{\it \_F1} \left ( {\frac {ay-b\int \!{x}^{\beta \,x}{{\rm e}^{-x\lambda }}\,{\rm d}x}{a}} \right ) \right ) {{\rm e}^{{\frac {c{{\rm e}^{ \left ( \gamma -\lambda \right ) x}}}{ \left ( \gamma -\lambda \right ) a}}}}\]
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