48 HFOPDE, chapter 2.4.4

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48.1 problem number 1

problem number 412

Added January 10, 2019.

Problem 2.4.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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48.2 problem number 2

problem number 413

Added January 10, 2019.

Problem 2.4.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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48.3 problem number 3

problem number 414

Added January 10, 2019.

Problem 2.4.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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)*coth(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'));
 

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

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48.4 problem number 4

problem number 415

Added January 10, 2019.

Problem 2.4.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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)*Coth[lambda*x]^2)*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 + a*lambda  -lambda^2 - a*(a+lambda)*coth(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 ( {\rm coth} \left (\lambda \,x\right )\LegendreP \left ( {\frac {a}{\lambda }},{\frac {\sqrt {{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) a+{\rm coth} \left (\lambda \,x\right )\LegendreP \left ( {\frac {a}{\lambda }},{\frac {\sqrt {{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) \lambda +\LegendreP \left ( {\frac {a+\lambda }{\lambda }},{\frac {\sqrt {{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) \sqrt {{a}^{2}+{\lambda }^{2}}-\LegendreP \left ( {\frac {a+\lambda }{\lambda }},{\frac {\sqrt {{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) a-\LegendreP \left ( {\frac {a+\lambda }{\lambda }},{\frac {\sqrt {{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) \lambda +\LegendreP \left ( {\frac {a}{\lambda }},{\frac {\sqrt {{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) y \right ) \left ( \LegendreQ \left ( {\frac {a}{\lambda }},{\frac {\sqrt {{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) {\rm coth} \left (\lambda \,x\right )a+\LegendreQ \left ( {\frac {a}{\lambda }},{\frac {\sqrt {{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) {\rm coth} \left (\lambda \,x\right )\lambda +\LegendreQ \left ( {\frac {a+\lambda }{\lambda }},{\frac {\sqrt {{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) \sqrt {{a}^{2}+{\lambda }^{2}}-\LegendreQ \left ( {\frac {a+\lambda }{\lambda }},{\frac {\sqrt {{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) a-\LegendreQ \left ( {\frac {a+\lambda }{\lambda }},{\frac {\sqrt {{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) \lambda +\LegendreQ \left ( {\frac {a}{\lambda }},{\frac {\sqrt {{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) y \right ) ^{-1}} \right ) \]