6.5.5 3.1

6.5.5.1 [1231] Problem 1

problem number 1231

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'));
 

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

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6.5.5.2 [1232] Problem 2

problem number 1232

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'));
 

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

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6.5.5.3 [1233] Problem 3

problem number 1233

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'));
 

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

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6.5.5.4 [1234] Problem 4

problem number 1234

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'));
 

\[\text {Expression too large to display}\]

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6.5.5.5 [1235] Problem 5

problem number 1235

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'));
 

\[\text {Expression too large to display}\]

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6.5.5.6 [1236] Problem 6

problem number 1236

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]];
 

\[\left \{\left \{w(x,y)\to \exp \left (\frac {c (\gamma -\beta ) \left (e^{\lambda y}\right )^{\frac {\sigma }{\lambda }} e^{-\gamma x-\lambda y} \operatorname {Hypergeometric2F1}\left (1,-\frac {\gamma }{\beta -\gamma },\frac {\beta \sigma -\gamma (\lambda +\sigma )}{(\beta -\gamma ) \lambda },1-\frac {a e^{\beta x-\gamma x-\lambda y} (\beta -\gamma )}{b \lambda }\right )}{b (\beta (\lambda -\sigma )+\gamma \sigma )}\right ) \left (\int _1^x\frac {\exp \left (\frac {c e^{-\gamma K[1]} (\beta -\gamma ) \left (\frac {a e^{\lambda y+\beta (x+K[1])} (\beta -\gamma )}{a e^{\beta (x+K[1])} (\beta -\gamma )-b e^{\lambda y} \left (e^{\gamma x+\beta K[1]}-e^{\beta x+\gamma K[1]}\right ) \lambda }\right )^{\frac {\sigma }{\lambda }-1} \operatorname {Hypergeometric2F1}\left (1,-\frac {\gamma }{\beta -\gamma },\frac {\beta \sigma -\gamma (\lambda +\sigma )}{(\beta -\gamma ) \lambda },e^{(\beta -\gamma ) K[1]} \left (\frac {a e^{-\lambda y} (\gamma -\beta )}{b \lambda }+e^{(\gamma -\beta ) x}\right )\right )}{b (\beta (\lambda -\sigma )+\gamma \sigma )}-\beta K[1]\right ) \left (e^{\mu K[1]} k \left (\frac {a e^{\lambda y+\beta (x+K[1])} (\gamma -\beta )}{b e^{\lambda y} \left (e^{\gamma x+\beta K[1]}-e^{\beta x+\gamma K[1]}\right ) \lambda -a e^{\beta (x+K[1])} (\beta -\gamma )}\right )^{\delta /\lambda }+d\right )}{a}dK[1]+c_1\left (\frac {b e^{\gamma x-\beta x}}{a \beta -a \gamma }-\frac {e^{-\lambda y}}{\lambda }\right )\right )\right \}\right \}\]

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'));
 

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

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6.5.5.7 [1237] Problem 7

problem number 1237

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'));
 

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

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6.5.5.8 [1238] Problem 8

problem number 1238

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'));
 

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

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