6.3.6 3.1

6.3.6.1 [877] Problem 1

problem number 877

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'));
 

\[w \left (x , y\right ) = \frac {f_{1} \left (\frac {y a -b x}{a}\right ) \mu b a \lambda +d \,{\mathrm e}^{\mu y} a \lambda +c \,{\mathrm e}^{\lambda x} \mu b}{\mu b a \lambda }\]

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6.3.6.2 [878] Problem 2

problem number 878

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'));
 

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

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6.3.6.3 [879] Problem 3

problem number 879

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'));
 

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

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6.3.6.4 [880] Problem 4

problem number 880

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'));
 

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

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6.3.6.5 [881] Problem 5

problem number 881

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'));
 

\[w \left (x , y\right ) = \frac {2 a^{2} \left (\alpha -\frac {\gamma }{2}\right ) \left (\alpha -\gamma \right ) f_{1} \left (\frac {{\mathrm e}^{-\alpha x} \beta b -{\mathrm e}^{-\beta y} a \alpha }{\alpha b \beta }\right )-2 \,{\mathrm e}^{\gamma x} c \left (a \left (\alpha -\frac {\gamma }{2}\right ) {\mathrm e}^{-\alpha x -\beta y}-\frac {{\mathrm e}^{-2 \alpha x} b \beta }{2}\right )}{a^{2} \left (2 \alpha -\gamma \right ) \left (\alpha -\gamma \right )}\]

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6.3.6.6 [882] Problem 6

problem number 882

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'));
 

\[w \left (x , y\right ) = \frac {c \,b^{2} \beta ^{2} {\mathrm e}^{\gamma x} \left (-\frac {{\mathrm e}^{-3 \alpha x}}{3 \alpha -\gamma }-\frac {\left ({\mathrm e}^{-\alpha x} \beta b -{\mathrm e}^{-\beta y} a \alpha \right )^{2} {\mathrm e}^{-\alpha x}}{b^{2} \beta ^{2} \left (\alpha -\gamma \right )}+\frac {-2 \,{\mathrm e}^{-2 \alpha x -\beta y} a \alpha +2 \,{\mathrm e}^{-3 \alpha x} b \beta }{b \beta \left (2 \alpha -\gamma \right )}\right )}{a^{3} \alpha ^{2}}+f_{1} \left (\frac {{\mathrm e}^{-\alpha x} \beta b -{\mathrm e}^{-\beta y} a \alpha }{\alpha b \beta }\right )\]

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6.3.6.7 [883] Problem 7

problem number 883

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'));
 

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

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6.3.6.8 [884] Problem 8

problem number 884

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 _{}^{y}\frac {s \,{\mathrm e}^{\frac {\mu \operatorname {RootOf}\left (-\int {\mathrm e}^{-\frac {\beta ^{2} x a -\beta x a \gamma +b \lambda \,{\mathrm e}^{-\textit {\_a} \left (\beta -\gamma \right )}}{\left (\beta -\gamma \right ) a}}d x \beta c \lambda -c \,{\mathrm e}^{\frac {-b \lambda \,{\mathrm e}^{-\textit {\_a} \left (\beta -\gamma \right )}+\textit {\_Z} a \beta }{\left (\beta -\gamma \right ) a}} \lambda -{\mathrm e}^{-\frac {\lambda \left (y a \beta -y a \gamma +{\mathrm e}^{-x \left (\beta -\gamma \right )} b \right )}{\left (\beta -\gamma \right ) a}} a \beta +{\mathrm e}^{-\frac {\lambda \left (\textit {\_b} a \beta -a \textit {\_b} \gamma +b \,{\mathrm e}^{\textit {\_Z}}\right )}{\left (\beta -\gamma \right ) a}} a \beta \right )}{-\beta +\gamma }}+k \,{\mathrm e}^{\delta \textit {\_b}}+p}{b \,{\mathrm e}^{-\frac {\gamma \operatorname {RootOf}\left (-\int {\mathrm e}^{-\frac {\beta ^{2} x a -\beta x a \gamma +b \lambda \,{\mathrm e}^{-\textit {\_a} \left (\beta -\gamma \right )}}{\left (\beta -\gamma \right ) a}}d x \beta c \lambda -c \,{\mathrm e}^{\frac {-b \lambda \,{\mathrm e}^{-\textit {\_a} \left (\beta -\gamma \right )}+\textit {\_Z} a \beta }{\left (\beta -\gamma \right ) a}} \lambda -{\mathrm e}^{-\frac {\lambda \left (y a \beta -y a \gamma +{\mathrm e}^{-x \left (\beta -\gamma \right )} b \right )}{\left (\beta -\gamma \right ) a}} a \beta +{\mathrm e}^{-\frac {\lambda \left (\textit {\_b} a \beta -a \textit {\_b} \gamma +b \,{\mathrm e}^{\textit {\_Z}}\right )}{\left (\beta -\gamma \right ) a}} a \beta \right )}{\beta -\gamma }}+c \,{\mathrm e}^{\lambda \textit {\_b}}}d \textit {\_b} +f_{1} \left (\frac {\frac {c \,{\mathrm e}^{\frac {-b \lambda \,{\mathrm e}^{-\textit {\_a} \left (\beta -\gamma \right )}-x a \beta \left (\beta -\gamma \right )}{\left (\beta -\gamma \right ) a}} \lambda }{a \beta }-{\mathrm e}^{-\frac {\left ({\mathrm e}^{-x \left (\beta -\gamma \right )} b +y \left (\beta -\gamma \right ) a \right ) \lambda }{\left (\beta -\gamma \right ) a}}}{\lambda }\right )\]

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6.3.6.9 [885] Problem 9

problem number 885

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 _{}^{y}\frac {s \,{\mathrm e}^{\frac {\mu \operatorname {RootOf}\left (-\int {\mathrm e}^{-\frac {\beta ^{2} x a -\beta x a \gamma +\lambda b \,{\mathrm e}^{-\textit {\_a} \left (\beta -\gamma \right )}}{\left (\beta -\gamma \right ) a}}d x \beta c \lambda -c \,{\mathrm e}^{\frac {-\lambda b \,{\mathrm e}^{-\textit {\_a} \left (\beta -\gamma \right )}+\textit {\_Z} a \beta }{\left (\beta -\gamma \right ) a}} \lambda +{\mathrm e}^{-\frac {\lambda \left (\textit {\_b} a \beta -\textit {\_b} a \gamma +b \,{\mathrm e}^{\textit {\_Z}}\right )}{\left (\beta -\gamma \right ) a}} a \beta -{\mathrm e}^{-\frac {\lambda \left (y a \beta -y a \gamma +b \,{\mathrm e}^{-x \left (\beta -\gamma \right )}\right )}{\left (\beta -\gamma \right ) a}} a \beta \right )+\delta \textit {\_b} \left (-\beta +\gamma \right )}{-\beta +\gamma }}+k}{b \,{\mathrm e}^{-\frac {\gamma \operatorname {RootOf}\left (-\int {\mathrm e}^{-\frac {\beta ^{2} x a -\beta x a \gamma +\lambda b \,{\mathrm e}^{-\textit {\_a} \left (\beta -\gamma \right )}}{\left (\beta -\gamma \right ) a}}d x \beta c \lambda -c \,{\mathrm e}^{\frac {-\lambda b \,{\mathrm e}^{-\textit {\_a} \left (\beta -\gamma \right )}+\textit {\_Z} a \beta }{\left (\beta -\gamma \right ) a}} \lambda +{\mathrm e}^{-\frac {\lambda \left (\textit {\_b} a \beta -\textit {\_b} a \gamma +b \,{\mathrm e}^{\textit {\_Z}}\right )}{\left (\beta -\gamma \right ) a}} a \beta -{\mathrm e}^{-\frac {\lambda \left (y a \beta -y a \gamma +b \,{\mathrm e}^{-x \left (\beta -\gamma \right )}\right )}{\left (\beta -\gamma \right ) a}} a \beta \right )}{\beta -\gamma }}+c \,{\mathrm e}^{\lambda \textit {\_b}}}d \textit {\_b} +f_{1} \left (\frac {-{\mathrm e}^{-\frac {\left (b \,{\mathrm e}^{-x \left (\beta -\gamma \right )}+y \left (\beta -\gamma \right ) a \right ) \lambda }{\left (\beta -\gamma \right ) a}}+\frac {c \,{\mathrm e}^{\frac {-\lambda b \,{\mathrm e}^{-\textit {\_a} \left (\beta -\gamma \right )}-x a \beta \left (\beta -\gamma \right )}{\left (\beta -\gamma \right ) a}} \lambda }{a \beta }}{\lambda }\right )\]

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6.3.6.10 [886] Problem 10

problem number 886

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]];
 

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

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'));
 

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

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6.3.6.11 [887] Problem 11

problem number 887

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]];
 

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

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'));
 

\[w \left (x , y\right ) = \beta \int _{}^{x}\frac {c \left (\frac {{\mathrm e}^{\lambda y} a \beta -\lambda b \left ({\mathrm e}^{\beta x}-{\mathrm e}^{\beta \textit {\_a}}\right )}{a \beta }\right )^{\frac {\gamma }{\lambda }}+d}{{\mathrm e}^{\lambda y} a \beta -\lambda b \left ({\mathrm e}^{\beta x}-{\mathrm e}^{\beta \textit {\_a}}\right )}d \textit {\_a} +f_{1} \left (\frac {{\mathrm e}^{\lambda y} a \beta -{\mathrm e}^{\beta x} \lambda b}{\beta \lambda b}\right )\]

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