6.4.6 3.1

6.4.6.1 [1059] Problem 1

problem number 1059

Added Feb. 23, 2019.

Problem Chapter 4.3.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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]; 
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

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

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

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6.4.6.2 [1060] Problem 2

problem number 1060

Added Feb. 23, 2019.

Problem Chapter 4.3.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.6.3 [1061] Problem 3

problem number 1061

Added Feb. 23, 2019.

Problem Chapter 4.3.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.6.4 [1062] Problem 4

problem number 1062

Added Feb. 23, 2019.

Problem Chapter 4.3.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.6.5 [1063] Problem 5

problem number 1063

Added Feb. 23, 2019.

Problem Chapter 4.3.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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

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

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6.4.6.6 [1064] Problem 6

problem number 1064

Added Feb. 23, 2019.

Problem Chapter 4.3.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.6.7 [1065] Problem 7

problem number 1065

Added Feb. 23, 2019.

Problem Chapter 4.3.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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6.4.6.8 [1066] Problem 8

problem number 1066

Added Feb. 23, 2019.

Problem Chapter 4.3.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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6.4.6.9 [1067] Problem 9

problem number 1067

Added Feb. 23, 2019.

Problem Chapter 4.3.1.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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

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

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6.4.6.10 [1068] Problem 10

problem number 1068

Added Feb. 23, 2019.

Problem Chapter 4.3.1.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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