75 HFOPDE, chapter 3.3.1

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75.1 Problem 1

problem number 686

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p]; 
 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]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
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'));
 

\[ w \left ( x,y \right ) ={\frac {1}{b\mu \,a\lambda } \left ( {\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) b\mu \,a\lambda +{{\rm e}^{\lambda \,x}}cb\mu +d{{\rm e}^{{\frac { \left ( ya-bx \right ) \mu }{a}}+{\frac {b\mu \,x}{a}}}}a\lambda \right ) } \]

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75.2 Problem 2

problem number 687

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p]; 
 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]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
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'));
 

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

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75.3 Problem 3

problem number 688

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p]; 
 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]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
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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75.4 Problem 4

problem number 689

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p]; 
 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]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
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'));
 

\[ w \left ( x,y \right ) ={\frac {1}{{{\rm e}^{y\lambda }}a\beta -{{\rm e}^{\beta \,x}}b\lambda } \left ( {{\rm e}^{y\lambda }}{\it \_F1} \left ( -{\frac {{{\rm e}^{\beta \,x}}b\lambda -{{\rm e}^{y\lambda }}a\beta }{b\beta \,\lambda }} \right ) a\beta -{\it \_F1} \left ( -{\frac {{{\rm e}^{\beta \,x}}b\lambda -{{\rm e}^{y\lambda }}a\beta }{b\beta \,\lambda }} \right ) {{\rm e}^{\beta \,x}}b\lambda -c\ln \left ( -{\frac {{{\rm e}^{\beta \,x}}b\lambda -{{\rm e}^{y\lambda }}a\beta }{\lambda \,b}}+{{\rm e}^{\beta \,x}} \right ) +c\ln \left ( {{\rm e}^{\beta \,x}} \right ) \right ) } \]

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75.5 Problem 5

problem number 690

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p]; 
 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]];
 

\[ \left \{\left \{w(x,y)\to \frac {e^{-\beta y} \left (2 a^2 \alpha ^2 e^{\beta y} c_1\left (-\frac {e^{-\alpha x-\beta y} \left (a \alpha e^{\alpha x}-b \beta e^{\beta y}\right )}{a \alpha \beta }\right )+a^2 \gamma ^2 e^{\beta y} c_1\left (-\frac {e^{-\alpha x-\beta y} \left (a \alpha e^{\alpha x}-b \beta e^{\beta y}\right )}{a \alpha \beta }\right )-3 a^2 \alpha \gamma e^{\beta y} c_1\left (-\frac {e^{-\alpha x-\beta y} \left (a \alpha e^{\alpha x}-b \beta e^{\beta y}\right )}{a \alpha \beta }\right )-2 a \alpha c e^{x (\gamma -2 \alpha )+\alpha x}+a c \gamma e^{x (\gamma -2 \alpha )+\alpha x}+b \beta c e^{x (\gamma -2 \alpha )+\beta y}\right )}{a^2 (\alpha -\gamma ) (2 \alpha -\gamma )}\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
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'));
 

\[ w \left ( x,y \right ) ={\frac {b\beta \,c}{{a}^{2}\alpha } \left ( -{\frac { \left ( {{\rm e}^{\beta \,y}}b\beta -a\alpha \,{{\rm e}^{\alpha \,x}} \right ) {{\rm e}^{-\alpha \,x-\beta \,y+x \left ( \gamma -\alpha \right ) }}}{ \left ( \gamma -\alpha \right ) b\beta }}+{\frac {{{\rm e}^{x \left ( \gamma -2\,\alpha \right ) }}}{\gamma -2\,\alpha }} \right ) }+{\it \_F1} \left ( {\frac { \left ( {{\rm e}^{\beta \,y}}b\beta -a\alpha \,{{\rm e}^{\alpha \,x}} \right ) {{\rm e}^{-\alpha \,x-\beta \,y}}}{\alpha \,b\beta }} \right ) \]

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75.6 Problem 6

problem number 691

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p]; 
 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]];
 

\[ \left \{\left \{w(x,y)\to \frac {e^{-2 \beta y} \left (6 a^3 \alpha ^3 e^{2 \beta y} c_1\left (-\frac {e^{-\alpha x-\beta y} \left (a \alpha e^{\alpha x}-b \beta e^{\beta y}\right )}{a \alpha \beta }\right )-11 a^3 \alpha ^2 \gamma e^{2 \beta y} c_1\left (-\frac {e^{-\alpha x-\beta y} \left (a \alpha e^{\alpha x}-b \beta e^{\beta y}\right )}{a \alpha \beta }\right )-a^3 \gamma ^3 e^{2 \beta y} c_1\left (-\frac {e^{-\alpha x-\beta y} \left (a \alpha e^{\alpha x}-b \beta e^{\beta y}\right )}{a \alpha \beta }\right )+6 a^3 \alpha \gamma ^2 e^{2 \beta y} c_1\left (-\frac {e^{-\alpha x-\beta y} \left (a \alpha e^{\alpha x}-b \beta e^{\beta y}\right )}{a \alpha \beta }\right )-6 a^2 \alpha ^2 c e^{x (\gamma -3 \alpha )+2 \alpha x}-a^2 c \gamma ^2 e^{x (\gamma -3 \alpha )+2 \alpha x}+5 a^2 \alpha c \gamma e^{x (\gamma -3 \alpha )+2 \alpha x}+6 a \alpha b \beta c e^{x (\gamma -3 \alpha )+\alpha x+\beta y}-2 a b \beta c \gamma e^{x (\gamma -3 \alpha )+\alpha x+\beta y}-2 b^2 \beta ^2 c e^{x (\gamma -3 \alpha )+2 \beta y}\right )}{a^3 (\alpha -\gamma ) (2 \alpha -\gamma ) (3 \alpha -\gamma )}\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
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'));
 

\[ w \left ( x,y \right ) =-{\frac {{b}^{2}{\beta }^{2}c}{{a}^{3}{\alpha }^{2}} \left ( 2\,{\frac { \left ( {{\rm e}^{\beta \,y}}b\beta -a\alpha \,{{\rm e}^{\alpha \,x}} \right ) {{\rm e}^{-\alpha \,x-\beta \,y+x \left ( \gamma -2\,\alpha \right ) }}}{b\beta \, \left ( \gamma -2\,\alpha \right ) }}-{\frac { \left ( {{\rm e}^{\beta \,y}}b\beta -a\alpha \,{{\rm e}^{\alpha \,x}} \right ) ^{2}{{\rm e}^{-2\,\alpha \,x-2\,\beta \,y+x \left ( \gamma -\alpha \right ) }}}{ \left ( \gamma -\alpha \right ) {b}^{2}{\beta }^{2}}}-{\frac {{{\rm e}^{x \left ( \gamma -3\,\alpha \right ) }}}{\gamma -3\,\alpha }} \right ) }+{\it \_F1} \left ( {\frac { \left ( {{\rm e}^{\beta \,y}}b\beta -a\alpha \,{{\rm e}^{\alpha \,x}} \right ) {{\rm e}^{-\alpha \,x-\beta \,y}}}{\alpha \,b\beta }} \right ) \]

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75.7 Problem 7

problem number 692

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p]; 
 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 \frac {e^{-\beta y} \left (\frac {e^{-\alpha x-\beta y} \left (a \alpha e^{\alpha x}-b \beta e^{\beta y}\right )}{a \alpha }+\frac {b \beta e^{-\alpha x}}{a \alpha }\right )^{-\frac {\mu }{\beta }} \left (-b \beta c e^{x (\gamma -\alpha )+\beta y} \left (\frac {e^{-\alpha x-\beta y} \left (a \alpha e^{\alpha x}-b \beta e^{\beta y}\right )}{a \alpha }+\frac {b \beta e^{-\alpha x}}{a \alpha }\right )^{\frac {\mu }{\beta }}+b c \mu e^{x (\gamma -\alpha )+\beta y} \left (\frac {e^{-\alpha x-\beta y} \left (a \alpha e^{\alpha x}-b \beta e^{\beta y}\right )}{a \alpha }+\frac {b \beta e^{-\alpha x}}{a \alpha }\right )^{\frac {\mu }{\beta }}-a b \beta \gamma e^{\beta y} c_1\left (-\frac {e^{-\alpha x-\beta y} \left (a \alpha e^{\alpha x}-b \beta e^{\beta y}\right )}{a \alpha \beta }\right ) \left (\frac {e^{-\alpha x-\beta y} \left (a \alpha e^{\alpha x}-b \beta e^{\beta y}\right )}{a \alpha }+\frac {b \beta e^{-\alpha x}}{a \alpha }\right )^{\frac {\mu }{\beta }}+a b \gamma \mu e^{\beta y} c_1\left (-\frac {e^{-\alpha x-\beta y} \left (a \alpha e^{\alpha x}-b \beta e^{\beta y}\right )}{a \alpha \beta }\right ) \left (\frac {e^{-\alpha x-\beta y} \left (a \alpha e^{\alpha x}-b \beta e^{\beta y}\right )}{a \alpha }+\frac {b \beta e^{-\alpha x}}{a \alpha }\right )^{\frac {\mu }{\beta }}+a \alpha b \beta e^{\beta y} c_1\left (-\frac {e^{-\alpha x-\beta y} \left (a \alpha e^{\alpha x}-b \beta e^{\beta y}\right )}{a \alpha \beta }\right ) \left (\frac {e^{-\alpha x-\beta y} \left (a \alpha e^{\alpha x}-b \beta e^{\beta y}\right )}{a \alpha }+\frac {b \beta e^{-\alpha x}}{a \alpha }\right )^{\frac {\mu }{\beta }}-a \alpha b \mu e^{\beta y} c_1\left (-\frac {e^{-\alpha x-\beta y} \left (a \alpha e^{\alpha x}-b \beta e^{\beta y}\right )}{a \alpha \beta }\right ) \left (\frac {e^{-\alpha x-\beta y} \left (a \alpha e^{\alpha x}-b \beta e^{\beta y}\right )}{a \alpha }+\frac {b \beta e^{-\alpha x}}{a \alpha }\right )^{\frac {\mu }{\beta }}-a \alpha s+a \gamma s\right )}{a b (\alpha -\gamma ) (\beta -\mu )}\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
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'));
 

\[ w \left ( x,y \right ) ={\frac {c{{\rm e}^{x \left ( \gamma -\alpha \right ) }}}{a \left ( \gamma -\alpha \right ) }}+{\frac {s \left ( {{\rm e}^{\beta \,y}}b\beta -a\alpha \,{{\rm e}^{\alpha \,x}} \right ) {{\rm e}^{-\alpha \,x-\beta \,y}}}{a\alpha \,b \left ( \beta -\mu \right ) } \left ( {\frac {a\alpha }{b\beta } \left ( -{\frac { \left ( {{\rm e}^{\beta \,y}}b\beta -a\alpha \,{{\rm e}^{\alpha \,x}} \right ) {{\rm e}^{-\alpha \,x-\beta \,y}}}{b\beta }}+{{\rm e}^{-\alpha \,x}} \right ) ^{-1}} \right ) ^{{\frac {\mu }{\beta }}}}-{\frac {\beta \,s{{\rm e}^{-\alpha \,x}}}{a\alpha \, \left ( \beta -\mu \right ) } \left ( {\frac {a\alpha }{b\beta } \left ( -{\frac { \left ( {{\rm e}^{\beta \,y}}b\beta -a\alpha \,{{\rm e}^{\alpha \,x}} \right ) {{\rm e}^{-\alpha \,x-\beta \,y}}}{b\beta }}+{{\rm e}^{-\alpha \,x}} \right ) ^{-1}} \right ) ^{{\frac {\mu }{\beta }}}}+{\it \_F1} \left ( {\frac { \left ( {{\rm e}^{\beta \,y}}b\beta -a\alpha \,{{\rm e}^{\alpha \,x}} \right ) {{\rm e}^{-\alpha \,x-\beta \,y}}}{\alpha \,b\beta }} \right ) \]

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75.8 Problem 8

problem number 693

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 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]];
 

\[ \text {Failed} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
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'));
 

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a} \left ( s{{\rm e}^{-{\it \_b}\, \left ( \beta -\mu \right ) }}+{{\rm e}^{-\beta \,{\it \_b}}}p+ \left ( -{\frac {\lambda }{a} \left ( c\int \!{{\rm e}^{{\frac {-a\beta \,\gamma \,{\it \_b}+a{\beta }^{2}{\it \_b}+\lambda \,b{{\rm e}^{{\it \_b}\, \left ( \gamma -\beta \right ) }}}{ \left ( \gamma -\beta \right ) a}}}}\,{\rm d}{\it \_b}-{\frac {a}{\lambda } \left ( \lambda \,\int \!{\frac {c}{a}{{\rm e}^{{\frac {-a\beta \,\gamma \,x+a{\beta }^{2}x+\lambda \,b{{\rm e}^{x \left ( \gamma -\beta \right ) }}}{ \left ( \gamma -\beta \right ) a}}}}}\,{\rm d}x+{{\rm e}^{-{\frac {\lambda \, \left ( -ya\beta +\gamma \,ya-b{{\rm e}^{x \left ( \gamma -\beta \right ) }} \right ) }{ \left ( \gamma -\beta \right ) a}}}} \right ) } \right ) } \right ) ^{-{\frac {\delta }{\lambda }}}k{{\rm e}^{{\frac {-a\beta \,\gamma \,{\it \_b}+a{\beta }^{2}{\it \_b}+\delta \,b{{\rm e}^{{\it \_b}\, \left ( \gamma -\beta \right ) }}}{ \left ( \gamma -\beta \right ) a}}}} \right ) }{d{\it \_b}}+{\it \_F1} \left ( -{\frac {1}{\lambda } \left ( \lambda \,\int \!{\frac {c}{a}{{\rm e}^{{\frac {-a\beta \,\gamma \,x+a{\beta }^{2}x+\lambda \,b{{\rm e}^{x \left ( \gamma -\beta \right ) }}}{ \left ( \gamma -\beta \right ) a}}}}}\,{\rm d}x+{{\rm e}^{-{\frac {\lambda \, \left ( -ya\beta +\gamma \,ya-b{{\rm e}^{x \left ( \gamma -\beta \right ) }} \right ) }{ \left ( \gamma -\beta \right ) a}}}} \right ) } \right ) \]

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75.9 Problem 9

problem number 694

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 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]];
 

\[ \text {Failed} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
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'));
 

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a} \left ( {{\rm e}^{-\beta \,{\it \_b}}}k+ \left ( -{\frac {\lambda }{a} \left ( c\int \!{{\rm e}^{{\frac {-a\beta \,\gamma \,{\it \_b}+a{\beta }^{2}{\it \_b}+\lambda \,b{{\rm e}^{{\it \_b}\, \left ( \gamma -\beta \right ) }}}{ \left ( \gamma -\beta \right ) a}}}}\,{\rm d}{\it \_b}-{\frac {a}{\lambda } \left ( \lambda \,\int \!{\frac {c}{a}{{\rm e}^{{\frac {-a\beta \,\gamma \,x+a{\beta }^{2}x+\lambda \,b{{\rm e}^{x \left ( \gamma -\beta \right ) }}}{ \left ( \gamma -\beta \right ) a}}}}}\,{\rm d}x+{{\rm e}^{-{\frac {\lambda \, \left ( -ya\beta +\gamma \,ya-b{{\rm e}^{x \left ( \gamma -\beta \right ) }} \right ) }{ \left ( \gamma -\beta \right ) a}}}} \right ) } \right ) } \right ) ^{-{\frac {\delta }{\lambda }}}s{{\rm e}^{{\frac {-a\beta \,\gamma \,{\it \_b}+{\it \_b}\,\mu \,a\gamma +a{\beta }^{2}{\it \_b}-{\it \_b}\,\mu \,a\beta +\delta \,b{{\rm e}^{{\it \_b}\, \left ( \gamma -\beta \right ) }}}{ \left ( \gamma -\beta \right ) a}}}} \right ) }{d{\it \_b}}+{\it \_F1} \left ( -{\frac {1}{\lambda } \left ( \lambda \,\int \!{\frac {c}{a}{{\rm e}^{{\frac {-a\beta \,\gamma \,x+a{\beta }^{2}x+\lambda \,b{{\rm e}^{x \left ( \gamma -\beta \right ) }}}{ \left ( \gamma -\beta \right ) a}}}}}\,{\rm d}x+{{\rm e}^{-{\frac {\lambda \, \left ( -ya\beta +\gamma \,ya-b{{\rm e}^{x \left ( \gamma -\beta \right ) }} \right ) }{ \left ( \gamma -\beta \right ) a}}}} \right ) } \right ) \]

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75.10 Problem 10

problem number 695

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 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]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
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'));
 

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a} \left ( c \left ( -{\frac { \left ( \gamma -\beta \right ) a}{\lambda \,b} \left ( {\frac { \left ( \lambda \,b{{\rm e}^{-\beta \,x+\gamma \,x+y\lambda }}+\gamma \,a-a\beta \right ) {{\rm e}^{-y\lambda }}\beta }{\lambda \,b \left ( \gamma -\beta \right ) }}-{\frac { \left ( \lambda \,b{{\rm e}^{-\beta \,x+\gamma \,x+y\lambda }}+\gamma \,a-a\beta \right ) {{\rm e}^{-y\lambda }}\gamma }{\lambda \,b \left ( \gamma -\beta \right ) }}+{{\rm e}^{{\it \_a}\, \left ( \gamma -\beta \right ) }} \right ) ^{-1}} \right ) ^{{\frac {\delta }{\lambda }}}{{\rm e}^{-{\it \_a}\, \left ( \beta -\mu \right ) }}+{{\rm e}^{-\beta \,{\it \_a}}}k \right ) }{d{\it \_a}}+{\it \_F1} \left ( -{\frac { \left ( \lambda \,b{{\rm e}^{-\beta \,x+\gamma \,x+y\lambda }}+\gamma \,a-a\beta \right ) {{\rm e}^{-y\lambda }}}{\lambda \,b \left ( \gamma -\beta \right ) }} \right ) \]

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75.11 Problem 11

problem number 696

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 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]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
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'));
 

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac {\beta }{\lambda \,b} \left ( c \left ( {\frac {\lambda \,b}{a\beta } \left ( -{\frac {{{\rm e}^{\beta \,x}}b\lambda -{{\rm e}^{y\lambda }}a\beta }{\lambda \,b}}+{{\rm e}^{\beta \,{\it \_a}}} \right ) } \right ) ^{{\frac {\gamma }{\lambda }}}+d \right ) \left ( -{\frac {{{\rm e}^{\beta \,x}}b\lambda -{{\rm e}^{y\lambda }}a\beta }{\lambda \,b}}+{{\rm e}^{\beta \,{\it \_a}}} \right ) ^{-1}}{d{\it \_a}}+{\it \_F1} \left ( -{\frac {{{\rm e}^{\beta \,x}}b\lambda -{{\rm e}^{y\lambda }}a\beta }{b\beta \,\lambda }} \right ) \]