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'));
\[w \left ( x,y \right ) ={\frac {1}{\mu \,ba\lambda } \left ( {\it \_F1} \left ( {\frac {ay-bx}{a}} \right ) \mu \,ba\lambda +d{{\rm e}^{\mu \,y}}a\lambda +c{{\rm e}^{\lambda \,x}}\mu \,b \right ) }\]
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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'));
\[w \left ( x,y \right ) ={\frac {c{{\rm e}^{\beta \,y+\lambda \,x}}}{a\lambda +b\beta }}+{\it \_F1} \left ( {\frac {ay-bx}{a}} \right ) \]
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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'));
\[w \left ( x,y \right ) ={\frac {1}{a\lambda } \left ( {\it \_F1} \left ( {\frac { \left ( {{\rm e}^{\beta \,y}}b\beta -a\lambda \,{{\rm e}^{\lambda \,x}} \right ) {{\rm e}^{-\beta \,y-\lambda \,x}}}{b\beta \,\lambda }} \right ) a\lambda -c{{\rm e}^{-\lambda \,x}} \right ) }\]
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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'));
\[w \left ( x,y \right ) ={\frac {1}{{{\rm e}^{y\lambda }}a\beta -{{\rm e}^{\beta \,x}}b\lambda } \left ( \left ( {{\rm e}^{y\lambda }}a\beta -{{\rm e}^{\beta \,x}}b\lambda \right ) {\it \_F1} \left ( {\frac {{{\rm e}^{y\lambda }}a\beta -{{\rm e}^{\beta \,x}}b\lambda }{b\beta \,\lambda }} \right ) +c \left ( \ln \left ( {{\rm e}^{\beta \,x}} \right ) -\ln \left ( {\frac {{{\rm e}^{y\lambda }}a\beta }{b\lambda }} \right ) \right ) \right ) }\]
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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'));
\[w \left ( x,y \right ) =-{\frac {b\beta \,c}{{a}^{2}\alpha } \left ( -{\frac { \left ( -{{\rm e}^{\beta \,y}}b\beta +a\alpha \,{{\rm e}^{\alpha \,x}} \right ) {{\rm e}^{x \left ( \gamma -2\,\alpha \right ) -\beta \,y}}}{b\beta \, \left ( \gamma -\alpha \right ) }}-{\frac {{{\rm e}^{x \left ( \gamma -2\,\alpha \right ) }}}{\gamma -2\,\alpha }} \right ) }+{\it \_F1} \left ( -{\frac { \left ( -{{\rm e}^{\beta \,y}}b\beta +a\alpha \,{{\rm e}^{\alpha \,x}} \right ) {{\rm e}^{-\alpha \,x-\beta \,y}}}{\alpha \,b\beta }} \right ) \]
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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]];
\[\left \{\left \{w(x,y)\to \frac {e^{-2 (\alpha x+\beta y)} \left (-c e^{x (\gamma -\alpha )} \left (a^2 \left (6 \alpha ^2-5 \alpha \gamma +\gamma ^2\right ) e^{2 \alpha x}-2 a b \beta (3 \alpha -\gamma ) e^{\alpha x+\beta y}+2 b^2 \beta ^2 e^{2 \beta y}\right )+a^3 \left (-11 \alpha ^2 \gamma +6 \alpha ^3+6 \alpha \gamma ^2-\gamma ^3\right ) e^{2 \alpha x+2 \beta y} c_1\left (\frac {b e^{-\alpha x}}{a \alpha }-\frac {e^{-\beta y}}{\beta }\right )\right )}{a^3 (\alpha -\gamma ) (2 \alpha -\gamma ) (3 \alpha -\gamma )}\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\frac {{b}^{2}{\beta }^{2}c}{{a}^{3}{\alpha }^{2}} \left ( {\frac { \left ( -{{\rm e}^{\beta \,y}}b\beta +a\alpha \,{{\rm e}^{\alpha \,x}} \right ) ^{2}{{\rm e}^{x \left ( \gamma -3\,\alpha \right ) -2\,\beta \,y}}}{{b}^{2}{\beta }^{2} \left ( \gamma -\alpha \right ) }}+2\,{\frac { \left ( -{{\rm e}^{\beta \,y}}b\beta +a\alpha \,{{\rm e}^{\alpha \,x}} \right ) {{\rm e}^{x \left ( \gamma -3\,\alpha \right ) -\beta \,y}}}{b\beta \, \left ( \gamma -2\,\alpha \right ) }}+{\frac {{{\rm e}^{x \left ( \gamma -3\,\alpha \right ) }}}{\gamma -3\,\alpha }} \right ) }+{\it \_F1} \left ( -{\frac { \left ( -{{\rm e}^{\beta \,y}}b\beta +a\alpha \,{{\rm e}^{\alpha \,x}} \right ) {{\rm e}^{-\alpha \,x-\beta \,y}}}{\alpha \,b\beta }} \right ) \]
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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'));
\[w \left ( x,y \right ) ={\frac {1}{a \left ( \gamma -\alpha \right ) \alpha \,b \left ( -\beta +\mu \right ) } \left ( s \left ( \gamma -\alpha \right ) \left ( \left ( -{{\rm e}^{\beta \,y}}b\beta +a\alpha \,{{\rm e}^{\alpha \,x}} \right ) {{\rm e}^{-\alpha \,x-\beta \,y}}+{{\rm e}^{-\alpha \,x}}\beta \,b \right ) \left ( {\frac {a\alpha }{ \left ( -{{\rm e}^{\beta \,y}}b\beta +a\alpha \,{{\rm e}^{\alpha \,x}} \right ) {{\rm e}^{-\alpha \,x-\beta \,y}}+{{\rm e}^{-\alpha \,x}}\beta \,b}} \right ) ^{{\frac {\mu }{\beta }}}+ \left ( a \left ( \gamma -\alpha \right ) {\it \_F1} \left ( -{\frac { \left ( -{{\rm e}^{\beta \,y}}b\beta +a\alpha \,{{\rm e}^{\alpha \,x}} \right ) {{\rm e}^{-\alpha \,x-\beta \,y}}}{b\alpha \,\beta }} \right ) +c{{\rm e}^{x \left ( \gamma -\alpha \right ) }} \right ) b\alpha \, \left ( -\beta +\mu \right ) \right ) }\]
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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'));
\[w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a} \left ( {{\rm e}^{-\beta \,{\it \_b}}}p+s{{\rm e}^{{\it \_b}\, \left ( -\beta +\mu \right ) }}+ \left ( {\frac {1}{a} \left ( a\int \!{\frac {c}{a}{{\rm e}^{{\frac {-\lambda \,b{{\rm e}^{x \left ( \gamma -\beta \right ) }}-ax\beta \, \left ( -\gamma +\beta \right ) }{a \left ( -\gamma +\beta \right ) }}}}}\,{\rm d}x\lambda -c\int \!{{\rm e}^{{\frac {-\lambda \,b{{\rm e}^{{\it \_b}\, \left ( \gamma -\beta \right ) }}-a{\it \_b}\,\beta \, \left ( -\gamma +\beta \right ) }{a \left ( -\gamma +\beta \right ) }}}}\,{\rm d}{\it \_b}\lambda +a{{\rm e}^{-{\frac { \left ( b{{\rm e}^{x \left ( \gamma -\beta \right ) }}+ay \left ( -\gamma +\beta \right ) \right ) \lambda }{a \left ( -\gamma +\beta \right ) }}}} \right ) } \right ) ^{-{\frac {\delta }{\lambda }}}k{{\rm e}^{{\frac {-\delta \,b{{\rm e}^{{\it \_b}\, \left ( \gamma -\beta \right ) }}-a{\it \_b}\,\beta \, \left ( -\gamma +\beta \right ) }{a \left ( -\gamma +\beta \right ) }}}} \right ) }{d{\it \_b}}+{\it \_F1} \left ( {\frac {1}{\lambda } \left ( -\lambda \,\int \!{\frac {c}{a}{{\rm e}^{{\frac {-\lambda \,b{{\rm e}^{x \left ( \gamma -\beta \right ) }}-ax\beta \, \left ( -\gamma +\beta \right ) }{a \left ( -\gamma +\beta \right ) }}}}}\,{\rm d}x-{{\rm e}^{-{\frac { \left ( b{{\rm e}^{x \left ( \gamma -\beta \right ) }}+ay \left ( -\gamma +\beta \right ) \right ) \lambda }{a \left ( -\gamma +\beta \right ) }}}} \right ) } \right ) \]
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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'));
\[w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a} \left ( {{\rm e}^{-\beta \,{\it \_b}}}k+ \left ( {\frac {1}{a} \left ( a\int \!{\frac {c}{a}{{\rm e}^{{\frac {-\lambda \,b{{\rm e}^{x \left ( \gamma -\beta \right ) }}-ax\beta \, \left ( -\gamma +\beta \right ) }{a \left ( -\gamma +\beta \right ) }}}}}\,{\rm d}x\lambda -c\int \!{{\rm e}^{{\frac {-\lambda \,b{{\rm e}^{{\it \_b}\, \left ( \gamma -\beta \right ) }}-a{\it \_b}\,\beta \, \left ( -\gamma +\beta \right ) }{a \left ( -\gamma +\beta \right ) }}}}\,{\rm d}{\it \_b}\lambda +a{{\rm e}^{-{\frac { \left ( b{{\rm e}^{x \left ( \gamma -\beta \right ) }}+ay \left ( -\gamma +\beta \right ) \right ) \lambda }{a \left ( -\gamma +\beta \right ) }}}} \right ) } \right ) ^{-{\frac {\delta }{\lambda }}}s{{\rm e}^{{\frac {-\delta \,b{{\rm e}^{{\it \_b}\, \left ( \gamma -\beta \right ) }}+a{\it \_b}\, \left ( -\gamma +\beta \right ) \left ( -\beta +\mu \right ) }{a \left ( -\gamma +\beta \right ) }}}} \right ) }{d{\it \_b}}+{\it \_F1} \left ( {\frac {1}{\lambda } \left ( -\lambda \,\int \!{\frac {c}{a}{{\rm e}^{{\frac {-\lambda \,b{{\rm e}^{x \left ( \gamma -\beta \right ) }}-ax\beta \, \left ( -\gamma +\beta \right ) }{a \left ( -\gamma +\beta \right ) }}}}}\,{\rm d}x-{{\rm e}^{-{\frac { \left ( b{{\rm e}^{x \left ( \gamma -\beta \right ) }}+ay \left ( -\gamma +\beta \right ) \right ) \lambda }{a \left ( -\gamma +\beta \right ) }}}} \right ) } \right ) \]
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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'));
\[w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a} \left ( c \left ( {\frac {a \left ( -\gamma +\beta \right ) }{-b\lambda \,{{\rm e}^{-y\lambda }}{{\rm e}^{x \left ( \gamma -\beta \right ) +y\lambda }}+\lambda \,b{{\rm e}^{{\it \_a}\, \left ( \gamma -\beta \right ) }}+a{{\rm e}^{-y\lambda }} \left ( -\gamma +\beta \right ) }} \right ) ^{{\frac {\delta }{\lambda }}}{{\rm e}^{{\it \_a}\, \left ( -\beta +\mu \right ) }}+{{\rm e}^{-{\it \_a}\,\beta }}k \right ) }{d{\it \_a}}+{\it \_F1} \left ( -{\frac { \left ( -b\lambda \,{{\rm e}^{x \left ( \gamma -\beta \right ) +y\lambda }}+a \left ( -\gamma +\beta \right ) \right ) {{\rm e}^{-y\lambda }}}{b\lambda \, \left ( -\gamma +\beta \right ) }} \right ) \]
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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'));
\[w \left ( x,y \right ) =\int ^{x}\!{\frac {\beta }{{{\rm e}^{y\lambda }}a\beta -b\lambda \, \left ( {{\rm e}^{\beta \,x}}-{{\rm e}^{{\it \_a}\,\beta }} \right ) } \left ( c \left ( {\frac {{{\rm e}^{y\lambda }}a\beta -b\lambda \, \left ( {{\rm e}^{\beta \,x}}-{{\rm e}^{{\it \_a}\,\beta }} \right ) }{a\beta }} \right ) ^{{\frac {\gamma }{\lambda }}}+d \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {{{\rm e}^{y\lambda }}a\beta -{{\rm e}^{\beta \,x}}b\lambda }{b\beta \,\lambda }} \right ) \]
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