45 HFOPDE, chapter 2.4.1

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45.1 problem number 1

problem number 389

Added January 10, 2019.

Problem 2.4.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*Sinh[lambda*x]*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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*sinh(lambda*x)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {\cosh \left ( \lambda \,x \right ) a-y\lambda }{\lambda }} \right ) \]

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45.2 problem number 2

problem number 390

Added January 10, 2019.

Problem 2.4.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \sinh (\mu y)w_y = 0 \]

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*Sinh[mu*y]*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {\log \left (\tanh \left (\frac {\mu y}{2}\right )\right )-a \mu x}{\mu }\right )\right \}\right \} \]

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*sinh(mu*y)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {x\mu \,a+2\,\arctanh \left ( {{\rm e}^{\mu \,y}} \right ) }{\mu \,a}} \right ) \]

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45.3 problem number 3

problem number 391

Added January 10, 2019.

Problem 2.4.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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^2 + a*lambda*Sinh[lambda*x] - a^2*Sinh[lambda*x]^2)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (-\frac {e^{-\frac {a e^{-\lambda x}}{\lambda }} \left (-2 y e^{\frac {a e^{-\lambda x}}{\lambda }+\lambda x} \int _1^{e^{\lambda x}} \frac {e^{\frac {a \left (K[1]^2-1\right )}{\lambda K[1]}}}{K[1]} \, dK[1]+a e^{\frac {a e^{-\lambda x}}{\lambda }} \int _1^{e^{\lambda x}} \frac {e^{\frac {a \left (K[1]^2-1\right )}{\lambda K[1]}}}{K[1]} \, dK[1]+a e^{\frac {a e^{-\lambda x}}{\lambda }+2 \lambda x} \int _1^{e^{\lambda x}} \frac {e^{\frac {a \left (K[1]^2-1\right )}{\lambda K[1]}}}{K[1]} \, dK[1]-2 \lambda e^{\frac {a e^{\lambda x}}{\lambda }+\lambda x}\right )}{a e^{2 \lambda x}+a-2 y e^{\lambda x}}\right )\right \}\right \} \]

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^2 + a*lambda*sinh(lambda*x) - a^2*sinh(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 ( {\sqrt {\sinh \left ( \lambda \,x \right ) +i} \left ( -2\,i \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}\cosh \left ( \lambda \,x \right ) \HeunC \left ( {\frac {4\,ia}{\lambda }},-1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) a-2\,i \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}\HeunC \left ( {\frac {4\,ia}{\lambda }},-1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) y+2\,i\sinh \left ( \lambda \,x \right ) \cosh \left ( \lambda \,x \right ) \HeunCPrime \left ( {\frac {4\,ia}{\lambda }},-1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) \lambda - \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}\cosh \left ( \lambda \,x \right ) \HeunCPrime \left ( {\frac {4\,ia}{\lambda }},-1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) \lambda +2\,i\cosh \left ( \lambda \,x \right ) \HeunC \left ( {\frac {4\,ia}{\lambda }},-1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) a-4\,\sinh \left ( \lambda \,x \right ) \cosh \left ( \lambda \,x \right ) \HeunC \left ( {\frac {4\,ia}{\lambda }},-1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) a+2\,i\HeunC \left ( {\frac {4\,ia}{\lambda }},-1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) y-4\,\sinh \left ( \lambda \,x \right ) \HeunC \left ( {\frac {4\,ia}{\lambda }},-1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) y+\lambda \,\cosh \left ( \lambda \,x \right ) \HeunCPrime \left ( {\frac {4\,ia}{\lambda }},-1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) \right ) \left ( i \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}\cosh \left ( \lambda \,x \right ) \HeunC \left ( {\frac {4\,ia}{\lambda }},1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) \lambda -i \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}\cosh \left ( \lambda \,x \right ) \HeunCPrime \left ( {\frac {4\,ia}{\lambda }},1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) \lambda +2\,i \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{3}\cosh \left ( \lambda \,x \right ) \HeunC \left ( {\frac {4\,ia}{\lambda }},1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) a+ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{3}\cosh \left ( \lambda \,x \right ) \HeunCPrime \left ( {\frac {4\,ia}{\lambda }},1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) \lambda +2\,\HeunC \left ( {\frac {4\,ia}{\lambda }},1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) y+2\, \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}\cosh \left ( \lambda \,x \right ) \HeunC \left ( {\frac {4\,ia}{\lambda }},1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) a+2\,\sinh \left ( \lambda \,x \right ) \cosh \left ( \lambda \,x \right ) \HeunC \left ( {\frac {4\,ia}{\lambda }},1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) \lambda +\sinh \left ( \lambda \,x \right ) \cosh \left ( \lambda \,x \right ) \HeunCPrime \left ( {\frac {4\,ia}{\lambda }},1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) \lambda +2\,i\sinh \left ( \lambda \,x \right ) \cosh \left ( \lambda \,x \right ) \HeunC \left ( {\frac {4\,ia}{\lambda }},1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) a+2\,\cosh \left ( \lambda \,x \right ) \HeunC \left ( {\frac {4\,ia}{\lambda }},1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) a-i\cosh \left ( \lambda \,x \right ) \HeunCPrime \left ( {\frac {4\,ia}{\lambda }},1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) \lambda +2\, \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}\HeunC \left ( {\frac {4\,ia}{\lambda }},1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) y+2\,i\sinh \left ( \lambda \,x \right ) \HeunC \left ( {\frac {4\,ia}{\lambda }},1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) y-i\cosh \left ( \lambda \,x \right ) \HeunC \left ( {\frac {4\,ia}{\lambda }},1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) \lambda +2\,i \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{3}\HeunC \left ( {\frac {4\,ia}{\lambda }},1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) y \right ) ^{-1}} \right ) \]

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45.4 problem number 4

problem number 392

Added January 10, 2019.

Problem 2.4.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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] + lambda*(Sinh[lambda*x]*y^2 - Sinh[lambda*x]^3)*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)+ lambda*(sinh(lambda*x)*y^2 - sinh(lambda*x)^3)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( -\cosh \left ( \lambda \,x \right ) +y \right ) \sqrt {\pi }}{\sqrt {\pi }\cosh \left ( \lambda \,x \right ) \erfi \left ( \cosh \left ( \lambda \,x \right ) \right ) -\sqrt {\pi }\erfi \left ( \cosh \left ( \lambda \,x \right ) \right ) y-2\,{{\rm e}^{ \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}}}} \right ) \]

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45.5 problem number 5

problem number 393

Added January 10, 2019.

Problem 2.4.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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] + ((a*Sinh[lambda*x]^2 - lambda)*y^2 - a*Sinh[lambda*x]^2 + lambda - a)*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)+ ((a*sinh(lambda*x)^2-lambda)*y^2 - a*sinh(lambda*x)^2 + lambda - a)*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 ( { \left ( -2\, \left ( \cosh \left ( 2\,\lambda \,x \right ) \right ) ^{2} \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}ya+4\,\cosh \left ( 2\,\lambda \,x \right ) \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}ya+ \left ( \cosh \left ( 2\,\lambda \,x \right ) \right ) ^{2}\sinh \left ( 2\,\lambda \,x \right ) a+2\, \left ( \cosh \left ( 2\,\lambda \,x \right ) \right ) ^{2}y\lambda -2\, \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}ay-2\,\cosh \left ( 2\,\lambda \,x \right ) \sinh \left ( 2\,\lambda \,x \right ) a-2\,\cosh \left ( 2\,\lambda \,x \right ) \sinh \left ( 2\,\lambda \,x \right ) \lambda -4\,\cosh \left ( 2\,\lambda \,x \right ) y\lambda +\sinh \left ( 2\,\lambda \,x \right ) a+2\,\lambda \,\sinh \left ( 2\,\lambda \,x \right ) +2\,y\lambda \right ) \sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1} \left ( -8\,\sinh \left ( 2\,\lambda \,x \right ) {{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}\sqrt {-1+\cosh \left ( 2\,\lambda \,x \right ) }{\lambda }^{2}-\sinh \left ( 2\,\lambda \,x \right ) \sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}\int \!2\,{\frac { \left ( \cosh \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cosh \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xa-2\,\sinh \left ( 2\,\lambda \,x \right ) \sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}\int \!2\,{\frac { \left ( \cosh \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cosh \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\lambda -2\,\sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}\int \!2\,{\frac { \left ( \cosh \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cosh \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xy\lambda - \left ( \cosh \left ( 2\,\lambda \,x \right ) \right ) ^{2}\sinh \left ( 2\,\lambda \,x \right ) \sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}\int \!2\,{\frac { \left ( \cosh \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cosh \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xa-2\, \left ( \cosh \left ( 2\,\lambda \,x \right ) \right ) ^{2}\sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}\int \!2\,{\frac { \left ( \cosh \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cosh \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xy\lambda +2\, \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}\sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}\int \!2\,{\frac { \left ( \cosh \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cosh \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xya+2\,\cosh \left ( 2\,\lambda \,x \right ) \sinh \left ( 2\,\lambda \,x \right ) \sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}\int \!2\,{\frac { \left ( \cosh \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cosh \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xa+2\,\cosh \left ( 2\,\lambda \,x \right ) \sinh \left ( 2\,\lambda \,x \right ) \sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}\int \!2\,{\frac { \left ( \cosh \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cosh \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\lambda +4\,\cosh \left ( 2\,\lambda \,x \right ) \sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}\int \!2\,{\frac { \left ( \cosh \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cosh \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xy\lambda -4\,\sinh \left ( 2\,\lambda \,x \right ) {{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}\sqrt {-1+\cosh \left ( 2\,\lambda \,x \right ) }a\lambda +2\, \left ( \cosh \left ( 2\,\lambda \,x \right ) \right ) ^{2} \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}\sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}\int \!2\,{\frac { \left ( \cosh \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cosh \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xya-4\,\cosh \left ( 2\,\lambda \,x \right ) \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}\sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}\int \!2\,{\frac { \left ( \cosh \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cosh \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xya+4\,\cosh \left ( 2\,\lambda \,x \right ) \sinh \left ( 2\,\lambda \,x \right ) {{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}\sqrt {-1+\cosh \left ( 2\,\lambda \,x \right ) }a\lambda \right ) ^{-1}} \right ) \]

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45.6 problem number 6

problem number 394

Added January 10, 2019.

Problem 2.4.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \sinh (\lambda x) w_x + a \left ( \sinh (\mu y)\right )w_y = 0 \]

Mathematica ✓

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

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {\log \left (\tanh \left (\frac {\mu y}{2}\right ) \sinh ^{-\frac {a \mu }{\lambda }}\left (\frac {\lambda x}{2}\right ) \cosh ^{\frac {a \mu }{\lambda }}\left (\frac {\lambda x}{2}\right )\right )}{\mu }\right )\right \}\right \} \]

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

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -2\,{\frac {-\arctanh \left ( {{\rm e}^{\lambda \,x}} \right ) \mu \,a+\arctanh \left ( {{\rm e}^{\mu \,y}} \right ) \lambda }{\lambda \,\mu \,a}} \right ) \]

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45.7 problem number 7

problem number 395

Added January 10, 2019.

Problem 2.4.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \sinh (\mu y) w_x + a \left ( \sinh (\lambda x)\right )w_y = 0 \]

Mathematica ✓

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

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

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

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {\cosh \left ( \lambda \,x \right ) \mu \,a-\cosh \left ( \mu \,y \right ) \lambda }{\lambda \,\mu \,a}} \right ) \]