47 HFOPDE, chapter 2.4.3

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47.1 problem number 1

problem number 404

Added January 10, 2019.

Problem 2.4.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \tanh (\lambda x) w_y = 0 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = D[w[x, y], x] + a*Tanh[lambda*x]*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {\lambda y-a \log (\cosh (\lambda x))}{\lambda }\right )\right \}\right \} \]

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := diff(w(x,y),x)+a*tanh(lambda*x)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( 1/2\,{\frac {\ln \left ( \tanh \left ( \lambda \,x \right ) -1 \right ) a+\ln \left ( \tanh \left ( \lambda \,x \right ) +1 \right ) a+2\,y\lambda }{\lambda }} \right ) \]

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47.2 problem number 2

problem number 405

Added January 10, 2019.

Problem 2.4.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \tanh (\lambda y) w_y = 0 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = D[w[x, y], x] + a*Tanh[lambda*y]*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {\log (\sinh (\lambda y))-a \lambda x}{\lambda }\right )\right \}\right \} \]

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := diff(w(x,y),x)+a*tanh(lambda*x)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( 1/2\,{\frac {\ln \left ( \tanh \left ( \lambda \,x \right ) -1 \right ) a+\ln \left ( \tanh \left ( \lambda \,x \right ) +1 \right ) a+2\,y\lambda }{\lambda }} \right ) \]

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47.3 problem number 3

problem number 406

Added January 10, 2019.

Problem 2.4.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( y^2+a \lambda - a (a+\lambda ) \tanh ^2(\lambda x) \right ) w_y = 0 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = D[w[x, y], x] + (y^2 + a*lambda - a*(a + lambda)*Tanh[lambda*x]^2)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (-\frac {\lambda \left (-y e^{2 \lambda x} \text {Hypergeometric2F1}\left (-\frac {2 a}{\lambda },-\frac {a}{\lambda },1-\frac {a}{\lambda },-e^{2 \lambda x}\right )-y \text {Hypergeometric2F1}\left (-\frac {2 a}{\lambda },-\frac {a}{\lambda },1-\frac {a}{\lambda },-e^{2 \lambda x}\right )+a e^{2 \lambda x} \text {Hypergeometric2F1}\left (-\frac {2 a}{\lambda },-\frac {a}{\lambda },1-\frac {a}{\lambda },-e^{2 \lambda x}\right )-a \text {Hypergeometric2F1}\left (-\frac {2 a}{\lambda },-\frac {a}{\lambda },1-\frac {a}{\lambda },-e^{2 \lambda x}\right )+2 a e^{2 \lambda x} \left (e^{2 \lambda x}+1\right )^{\frac {2 a}{\lambda }}+2 a \left (e^{2 \lambda x}+1\right )^{\frac {2 a}{\lambda }}\right )}{2 a \left (y e^{2 x (a+\lambda )}-a e^{2 x (a+\lambda )}+y e^{2 a x}+a e^{2 a x}\right )}\right )\right \}\right \} \]

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := diff(w(x,y),x)+( y^2+a*lambda - a*(a+lambda)*tanh(lambda*x)^2)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{1 \left ( \tanh \left ( \lambda \,x \right ) \LegendreP \left ( {\frac {a}{\lambda }},{\frac {a}{\lambda }},\tanh \left ( \lambda \,x \right ) \right ) a+\tanh \left ( \lambda \,x \right ) \LegendreP \left ( {\frac {a}{\lambda }},{\frac {a}{\lambda }},\tanh \left ( \lambda \,x \right ) \right ) \lambda -\LegendreP \left ( {\frac {a+\lambda }{\lambda }},{\frac {a}{\lambda }},\tanh \left ( \lambda \,x \right ) \right ) \lambda +\LegendreP \left ( {\frac {a}{\lambda }},{\frac {a}{\lambda }},\tanh \left ( \lambda \,x \right ) \right ) y \right ) \left ( \tanh \left ( \lambda \,x \right ) \LegendreQ \left ( {\frac {a}{\lambda }},{\frac {a}{\lambda }},\tanh \left ( \lambda \,x \right ) \right ) a+\tanh \left ( \lambda \,x \right ) \LegendreQ \left ( {\frac {a}{\lambda }},{\frac {a}{\lambda }},\tanh \left ( \lambda \,x \right ) \right ) \lambda -\lambda \,\LegendreQ \left ( {\frac {a+\lambda }{\lambda }},{\frac {a}{\lambda }},\tanh \left ( \lambda \,x \right ) \right ) +\LegendreQ \left ( {\frac {a}{\lambda }},{\frac {a}{\lambda }},\tanh \left ( \lambda \,x \right ) \right ) y \right ) ^{-1}} \right ) \]

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47.4 problem number 4

problem number 407

Added January 10, 2019.

Problem 2.4.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( y^2+3 a \lambda - \lambda ^2 -a(a+\lambda ) \tanh ^2(\lambda x) \right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = D[w[x, y], x] + (y^2 + 3*a*lambda - lambda^2 - a*(a + lambda)*Tanh[lambda*x])*D[w[x, y], y] == 0; 
 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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := diff(w(x,y),x)+( y^2+3*a*lambda - lambda^2 -a*(a+lambda)*tanh(lambda*x))*diff(w(x,y),y) = 0; 
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 ( \sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda \right ) \left ( a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}} \right ) }{ \left ( \sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda \right ) \left ( \sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda \right ) }{2}^{-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }}} \left ( -{\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\sinh \left ( \lambda \,x \right ) a\lambda \,\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+{\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\sinh \left ( \lambda \,x \right ) {a}^{2}\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+4\,i{\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) {\lambda }^{2}\sqrt {{a}^{2}+4\,a\lambda -{\lambda }^{2}}-2\,i{\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) {a}^{2}\sqrt {{a}^{2}+4\,a\lambda -{\lambda }^{2}}+{\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) {a}^{2}\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+4\,i{\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) y\lambda \,\sqrt {{a}^{2}+4\,a\lambda -{\lambda }^{2}}-{\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) a\lambda \,\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+{\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\sinh \left ( \lambda \,x \right ) ya\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-6\,i{\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) a\lambda \,\sqrt {{a}^{2}+4\,a\lambda -{\lambda }^{2}}+{\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) ya\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,i{\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) ya\sqrt {{a}^{2}+4\,a\lambda -{\lambda }^{2}}+2\,{\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\sinh \left ( \lambda \,x \right ) ya\lambda +2\,\sinh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}{a}^{2}\lambda -2\,\sinh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}a{\lambda }^{2}+2\,\cosh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}ya\lambda +2\,\cosh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}{a}^{2}\lambda -2\,\cosh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}a{\lambda }^{2}+2\,\cosh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}y{a}^{2}+6\,\cosh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}ya\lambda -4\,{\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) y{\lambda }^{2}+2\,\cosh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}{a}^{3}+12\,\cosh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}{a}^{2}\lambda +14\,\cosh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}a{\lambda }^{2}-4\,\cosh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}{\lambda }^{3} \right ) \left ( {\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }} \right ) ^{-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }}} \left ( 4\,i\cosh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}{\lambda }^{2}\sqrt {{a}^{2}+4\,a\lambda -{\lambda }^{2}}+4\,i\cosh \left ( \lambda \,x \right ) y{\mbox {$_2$F$_1$}(1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\lambda \,\sqrt {{a}^{2}+4\,a\lambda -{\lambda }^{2}}-\sinh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}a\lambda \,\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\cosh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}{a}^{2}\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\sinh \left ( \lambda \,x \right ) y{\mbox {$_2$F$_1$}(1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}a\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\sinh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}{a}^{2}\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\cosh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}a\lambda \,\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,i\cosh \left ( \lambda \,x \right ) y{\mbox {$_2$F$_1$}(1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}a\sqrt {{a}^{2}+4\,a\lambda -{\lambda }^{2}}-2\,i\cosh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}{a}^{2}\sqrt {{a}^{2}+4\,a\lambda -{\lambda }^{2}}+\cosh \left ( \lambda \,x \right ) y{\mbox {$_2$F$_1$}(1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}a\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-6\,i\cosh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}a\lambda \,\sqrt {{a}^{2}+4\,a\lambda -{\lambda }^{2}}-2\,\sinh \left ( \lambda \,x \right ) y{\mbox {$_2$F$_1$}(1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}a\lambda -2\,\sinh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}{a}^{2}\lambda +2\,\sinh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}a{\lambda }^{2}-2\,\cosh \left ( \lambda \,x \right ) y{\mbox {$_2$F$_1$}(1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}a\lambda -2\,\cosh \left ( \lambda \,x \right ) y{\mbox {$_2$F$_1$}(1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}{a}^{2}-6\,\cosh \left ( \lambda \,x \right ) y{\mbox {$_2$F$_1$}(1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}a\lambda +4\,\cosh \left ( \lambda \,x \right ) y{\mbox {$_2$F$_1$}(1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}{\lambda }^{2}-2\,\cosh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}{a}^{2}\lambda +2\,\cosh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}a{\lambda }^{2}-2\,\cosh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}{a}^{3}-12\,\cosh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}{a}^{2}\lambda -14\,\cosh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}a{\lambda }^{2}+4\,\cosh \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}{\lambda }^{3} \right ) ^{-1}} \right ) \]

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47.5 problem number 5

problem number 408

Added January 10, 2019.

Problem 2.4.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \left ( a x^n + b x \tanh ^m(y)\right ) w_x + y^k w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = (a*x^n + b*x*Tanh[y]^m)*D[w[x, y], x] + y^k*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {\$Aborted} \] Timed out

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := ( a*x^n + b*x*tanh(y)^m)*diff(w(x,y),x)+y^k*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {x}^{-n+1}{{\rm e}^{b\int \! \left ( \tanh \left ( y \right ) \right ) ^{m}{y}^{-k}\,{\rm d}y \left ( n-1 \right ) }}+an\int \!{{\rm e}^{b\int \! \left ( \tanh \left ( y \right ) \right ) ^{m}{y}^{-k}\,{\rm d}y \left ( n-1 \right ) }}{y}^{-k}\,{\rm d}y-a\int \!{{\rm e}^{b\int \! \left ( \tanh \left ( y \right ) \right ) ^{m}{y}^{-k}\,{\rm d}y \left ( n-1 \right ) }}{y}^{-k}\,{\rm d}y \right ) \]

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47.6 problem number 6

problem number 409

Added January 10, 2019.

Problem 2.4.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \left ( a x^n + b x \tanh ^m(y)\right ) w_x + \tanh ^k(\lambda y) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = (a*x^n + b*x*Tanh[y]^m)*D[w[x, y], x] + Tanh[lambda*y]^k*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {\$Aborted} \] Timed out

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := ( a*x^n + b*x*tanh(y)^m)*diff(w(x,y),x)+tanh(lambda*y)^k*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {x}^{-n+1}{{\rm e}^{b\int \! \left ( \tanh \left ( y \right ) \right ) ^{m} \left ( \tanh \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }}+an\int \!{{\rm e}^{b\int \! \left ( \tanh \left ( y \right ) \right ) ^{m} \left ( \tanh \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }} \left ( \tanh \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y-a\int \!{{\rm e}^{b\int \! \left ( \tanh \left ( y \right ) \right ) ^{m} \left ( \tanh \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }} \left ( \tanh \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \right ) \]

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47.7 problem number 7

problem number 410

Added January 10, 2019.

Problem 2.4.3.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \left ( a x^n y^m + b x\right ) w_x + \tanh ^k(\lambda y) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = (a*x^n*y^m + b*x)*D[w[x, y], x] + Tanh[lambda*y]^k*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {\$Aborted} \] Timed out

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := ( a*x^n*y^m + b*x)*diff(w(x,y),x)+tanh(lambda*y)^k*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {x}^{-n+1}{{\rm e}^{b\int \! \left ( \tanh \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }}+an\int \!{{\rm e}^{b\int \! \left ( \tanh \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }}{y}^{m} \left ( \tanh \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y-a\int \!{{\rm e}^{b\int \! \left ( \tanh \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }}{y}^{m} \left ( \tanh \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \right ) \]

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47.8 problem number 8

problem number 411

Added January 10, 2019.

Problem 2.4.3.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \left ( a x^n \tanh ^m y + b x\right ) w_x + y^k w_y = 0 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = (a*x^n*Tanh[y]^m)*D[w[x, y], x] + y^k*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {a n \int _1^y K[1]^{-k} \tanh ^m(K[1]) \, dK[1]-a \int _1^y K[1]^{-k} \tanh ^m(K[1]) \, dK[1]+x^{1-n}}{a n-a}\right )\right \}\right \} \]

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := ( a*x^n*tanh(y)^m)*diff(w(x,y),x)+y^k*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {x}^{-n+1}+an\int \! \left ( \tanh \left ( y \right ) \right ) ^{m}{y}^{-k}\,{\rm d}y-a\int \! \left ( \tanh \left ( y \right ) \right ) ^{m}{y}^{-k}\,{\rm d}y \right ) \]