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'));
 

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

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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'));
 

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

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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'));
 

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

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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'));
 

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

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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'));
 

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

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6.4.6.6 [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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {b e^{-\lambda x}}{a \lambda }-\frac {e^{-\beta y}}{\beta }\right ) \exp \left (-\frac {e^{-\lambda x} \left (e^{-\beta y}\right )^{-\frac {\delta +\gamma }{\beta }} \left ((\beta -\delta ) \left (c \gamma \left (e^{-\beta y}\right )^{\frac {\delta }{\beta }} \left (\frac {a \lambda e^{\lambda x-\beta y}}{b \beta }\right )^{\frac {\gamma }{\beta }} \, _2F_1\left (\frac {\beta +\gamma }{\beta },\frac {\gamma }{\beta }-1;\frac {\gamma }{\beta };1-\frac {a e^{\lambda x-\beta y} \lambda }{b \beta }\right )+(\beta -\gamma ) \left (c \left (e^{-\beta y}\right )^{\frac {\delta }{\beta }}+s \left (e^{-\beta y}\right )^{\frac {\gamma }{\beta }}\right )\right )+\delta s (\beta -\gamma ) \left (e^{-\beta y}\right )^{\frac {\gamma }{\beta }} \left (\frac {a \lambda e^{\lambda x-\beta y}}{b \beta }\right )^{\frac {\delta }{\beta }} \, _2F_1\left (\frac {\beta +\delta }{\beta },\frac {\delta }{\beta }-1;\frac {\delta }{\beta };1-\frac {a e^{\lambda x-\beta y} \lambda }{b \beta }\right )\right )}{a \lambda (\beta -\delta ) (\beta -\gamma )}\right )\right \}\right \}\]

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'));
 

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

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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'));
 

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

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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'));
 

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

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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]];
 

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

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'));
 

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

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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'));
 

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

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