46 HFOPDE, chapter 2.4.2

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46.1 problem number 1

problem number 396

Added January 10, 2019.

Problem 2.4.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \left ( \cosh (\lambda x)\right )w_y = 0 \]

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

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

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

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

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46.2 problem number 2

problem number 397

Added January 10, 2019.

Problem 2.4.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \left ( \cosh (\lambda x)\right )w_y = 0 \]

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

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {2 \tan ^{-1}\left (\tanh \left (\frac {\lambda y}{2}\right )\right )-a \lambda x}{\lambda }\right )\right \}\right \} \]

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

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

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46.3 problem number 3

problem number 398

Added January 10, 2019.

Problem 2.4.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*Cosh[lambda*x]^2 - lambda)*y^2 - a*Cosh[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 *cosh(lambda*x)^2-lambda)*y^2 - a*cosh(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 ( -{\sqrt {-1+\cosh \left ( 2\,\lambda \,x \right ) } \left ( -8\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{6}ya+8\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{4}y\lambda + \left ( \cosh \left ( 2\,\lambda \,x \right ) \right ) ^{2}\sinh \left ( 2\,\lambda \,x \right ) a+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 +\sinh \left ( 2\,\lambda \,x \right ) a-2\,\lambda \,\sinh \left ( 2\,\lambda \,x \right ) \right ) \left ( -8\,\sqrt {-1+\cosh \left ( 2\,\lambda \,x \right ) } \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{6}y\int \!2\,{\frac { \left ( \cosh \left ( 2\,\lambda \,x \right ) a+a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{\sqrt {-1+\cosh \left ( 2\,\lambda \,x \right ) } \left ( \cosh \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}}{{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xa+8\,\sqrt {-1+\cosh \left ( 2\,\lambda \,x \right ) } \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{4}y\int \!2\,{\frac { \left ( \cosh \left ( 2\,\lambda \,x \right ) a+a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{\sqrt {-1+\cosh \left ( 2\,\lambda \,x \right ) } \left ( \cosh \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}}{{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\lambda + \left ( \cosh \left ( 2\,\lambda \,x \right ) \right ) ^{2}\sinh \left ( 2\,\lambda \,x \right ) \sqrt {-1+\cosh \left ( 2\,\lambda \,x \right ) }\int \!2\,{\frac { \left ( \cosh \left ( 2\,\lambda \,x \right ) a+a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{\sqrt {-1+\cosh \left ( 2\,\lambda \,x \right ) } \left ( \cosh \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}}{{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xa-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 {\cosh \left ( 2\,\lambda \,x \right ) +1}a\lambda +2\,\cosh \left ( 2\,\lambda \,x \right ) \sinh \left ( 2\,\lambda \,x \right ) \sqrt {-1+\cosh \left ( 2\,\lambda \,x \right ) }\int \!2\,{\frac { \left ( \cosh \left ( 2\,\lambda \,x \right ) a+a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{\sqrt {-1+\cosh \left ( 2\,\lambda \,x \right ) } \left ( \cosh \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}}{{\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 {-1+\cosh \left ( 2\,\lambda \,x \right ) }\int \!2\,{\frac { \left ( \cosh \left ( 2\,\lambda \,x \right ) a+a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{\sqrt {-1+\cosh \left ( 2\,\lambda \,x \right ) } \left ( \cosh \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}}{{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\lambda -4\,\sinh \left ( 2\,\lambda \,x \right ) {{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}\sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}a\lambda +8\,\sinh \left ( 2\,\lambda \,x \right ) {{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}\sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}{\lambda }^{2}+\sinh \left ( 2\,\lambda \,x \right ) \sqrt {-1+\cosh \left ( 2\,\lambda \,x \right ) }\int \!2\,{\frac { \left ( \cosh \left ( 2\,\lambda \,x \right ) a+a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{\sqrt {-1+\cosh \left ( 2\,\lambda \,x \right ) } \left ( \cosh \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}}{{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xa-2\,\sinh \left ( 2\,\lambda \,x \right ) \sqrt {-1+\cosh \left ( 2\,\lambda \,x \right ) }\int \!2\,{\frac { \left ( \cosh \left ( 2\,\lambda \,x \right ) a+a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{\sqrt {-1+\cosh \left ( 2\,\lambda \,x \right ) } \left ( \cosh \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}}{{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\lambda \right ) ^{-1}} \right ) \]

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46.4 problem number 4

problem number 399

Added January 10, 2019.

Problem 2.4.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = 2*D[w[x, y], x] + ((a - lambda + a*Cosh[lambda*x])*y^2 + a + lambda - a*Cosh[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 := 2*diff(w(x,y),x)+ ( (a - lambda + a*cosh(lambda*x))*y^2 + a+ lambda- a *cosh(lambda*x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\sqrt {\cosh \left ( \lambda \,x \right ) -1} \left ( \cosh \left ( \lambda \,x \right ) +1 \right ) ^{3/2} \left ( -y\cosh \left ( \lambda \,x \right ) +\sinh \left ( \lambda \,x \right ) -y \right ) \left ( \sqrt {\cosh \left ( \lambda \,x \right ) -1} \left ( \cosh \left ( \lambda \,x \right ) +1 \right ) ^{5/2}\int \!{\frac { \left ( a-\lambda +\cosh \left ( \lambda \,x \right ) a \right ) \lambda \,\sinh \left ( \lambda \,x \right ) }{\sqrt {\cosh \left ( \lambda \,x \right ) -1} \left ( \cosh \left ( \lambda \,x \right ) +1 \right ) ^{3/2}}{{\rm e}^{{\frac {\cosh \left ( \lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xy-\sqrt {\cosh \left ( \lambda \,x \right ) -1} \left ( \cosh \left ( \lambda \,x \right ) +1 \right ) ^{3/2}\int \!{\frac { \left ( a-\lambda +\cosh \left ( \lambda \,x \right ) a \right ) \lambda \,\sinh \left ( \lambda \,x \right ) }{\sqrt {\cosh \left ( \lambda \,x \right ) -1} \left ( \cosh \left ( \lambda \,x \right ) +1 \right ) ^{3/2}}{{\rm e}^{{\frac {\cosh \left ( \lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\sinh \left ( \lambda \,x \right ) +2\,{{\rm e}^{{\frac {\cosh \left ( \lambda \,x \right ) a}{\lambda }}}}\sinh \left ( \lambda \,x \right ) \cosh \left ( \lambda \,x \right ) \lambda +2\,{{\rm e}^{{\frac {\cosh \left ( \lambda \,x \right ) a}{\lambda }}}}\sinh \left ( \lambda \,x \right ) \lambda \right ) ^{-1}} \right ) \]

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46.5 problem number 5

problem number 400

Added January 10, 2019.

Problem 2.4.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*Cosh[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*cosh(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'));
 

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

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46.6 problem number 6

problem number 401

Added January 10, 2019.

Problem 2.4.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*Cosh[y]^m)*D[w[x, y], x] + Cosh[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*cosh(y)^m)*diff(w(x,y),x)+cosh(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'));
 

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

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46.7 problem number 7

problem number 402

Added January 10, 2019.

Problem 2.4.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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] + Cosh[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)+cosh(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'));
 

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

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46.8 problem number 8

problem number 403

Added January 10, 2019.

Problem 2.4.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

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

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