103 HFOPDE, chapter 4.3.1

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103.1 Problem 1

problem number 868

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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]];
 

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

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';g1:='g1';g2:='g2';g0:='g0'; 
h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

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

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103.2 Problem 2

problem number 869

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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]];
 

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

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';g1:='g1';g2:='g2';g0:='g0'; 
h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

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

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103.3 Problem 3

problem number 870

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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]];
 

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

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';g1:='g1';g2:='g2';g0:='g0'; 
h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

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

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103.4 Problem 4

problem number 871

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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]];
 

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

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';g1:='g1';g2:='g2';g0:='g0'; 
h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

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

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103.5 Problem 5

problem number 872

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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]];
 

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

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';g1:='g1';g2:='g2';g0:='g0'; 
h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

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

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103.6 Problem 6

problem number 873

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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]];
 

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

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';g1:='g1';g2:='g2';g0:='g0'; 
h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

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

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103.7 Problem 7

problem number 874

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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]];
 

\[ \text {Failed} \]

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';g1:='g1';g2:='g2';g0:='g0'; 
h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

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

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103.8 Problem 8

problem number 875

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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]];
 

\[ \text {Failed} \]

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';g1:='g1';g2:='g2';g0:='g0'; 
h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

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

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103.9 Problem 9

problem number 876

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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]];
 

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

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';g1:='g1';g2:='g2';g0:='g0'; 
h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

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

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103.10 Problem 10

problem number 877

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 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]];
 

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

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';g1:='g1';g2:='g2';g0:='g0'; 
h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
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'));
 

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