6.2.8 4.1

6.2.8.1 [580] problem number 1

problem number 580

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["Global`*"]; 
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]];
 

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

Maple ✓

restart; 
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'));
 

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

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6.2.8.2 [581] problem number 2

problem number 581

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["Global`*"]; 
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]];
 

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

Maple ✓

restart; 
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'));
 

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

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6.2.8.3 [582] problem number 3

problem number 582

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["Global`*"]; 
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]];
 

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

Maple ✓

restart; 
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'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (-\frac {2 \sqrt {\sinh \left (\lambda x \right )+i}\, \left (-\left (-\frac {\left (\sinh ^{2}\left (\lambda x \right )\right )}{2}+i \sinh \left (\lambda x \right )+\frac {1}{2}\right ) \lambda \HeunCPrime \left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cosh \left (\lambda x \right )+\left (a \cosh \left (\lambda x \right )+y \right ) \left (i \left (\sinh ^{2}\left (\lambda x \right )\right )+2 \sinh \left (\lambda x \right )-i\right ) \HeunC \left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right )\right )}{-\left (\sinh ^{2}\left (\lambda x \right )+1\right ) \left (-\sinh \left (\lambda x \right )+i\right ) \lambda \HeunCPrime \left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cosh \left (\lambda x \right )+\left (2 \left (\sinh ^{2}\left (\lambda x \right )+1\right ) \left (i \sinh \left (\lambda x \right )+1\right ) y +\left (2 i a \left (\sinh ^{3}\left (\lambda x \right )\right )+\left (i \lambda +2 a \right ) \left (\sinh ^{2}\left (\lambda x \right )\right )+2 a -i \lambda +\left (2 i a +2 \lambda \right ) \sinh \left (\lambda x \right )\right ) \cosh \left (\lambda x \right )\right ) \HeunC \left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right )}\right )\]

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6.2.8.4 [583] problem number 4

problem number 583

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["Global`*"]; 
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]];
 

Failed

Maple ✓

restart; 
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'));
 

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

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6.2.8.5 [584] problem number 5

problem number 584

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["Global`*"]; 
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]];
 

Failed

Maple ✓

restart; 
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'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (-\frac {2 \left (\cosh \left (2 \lambda x \right )-1\right ) \left (-\left (a \left (\sinh ^{2}\left (\lambda x \right )\right )-\lambda \right ) y +\left (-\frac {a \sinh \left (2 \lambda x \right )}{2}+\left (a \left (\sinh ^{2}\left (\lambda x \right )\right )-\lambda \right ) y \right ) \cosh \left (2 \lambda x \right )+\left (\frac {a}{2}+\lambda \right ) \sinh \left (2 \lambda x \right )\right ) \sqrt {\cosh \left (2 \lambda x \right )+1}}{4 \sqrt {\cosh \left (2 \lambda x \right )-1}\, \left (a \cosh \left (2 \lambda x \right )-a -2 \lambda \right ) \lambda \,{\mathrm e}^{\frac {a \cosh \left (2 \lambda x \right )}{2 \lambda }} \sinh \left (2 \lambda x \right )+2 \left (\cosh \left (2 \lambda x \right )-1\right ) \left (-\left (a \left (\sinh ^{2}\left (\lambda x \right )\right )-\lambda \right ) y +\left (-\frac {a \sinh \left (2 \lambda x \right )}{2}+\left (a \left (\sinh ^{2}\left (\lambda x \right )\right )-\lambda \right ) y \right ) \cosh \left (2 \lambda x \right )+\left (\frac {a}{2}+\lambda \right ) \sinh \left (2 \lambda x \right )\right ) \sqrt {\cosh \left (2 \lambda x \right )+1}\, \left (\int \frac {2 \left (a \cosh \left (2 \lambda x \right )-a -2 \lambda \right ) \lambda \,{\mathrm e}^{\frac {a \cosh \left (2 \lambda x \right )}{2 \lambda }} \sinh \left (2 \lambda x \right )}{\left (\cosh \left (2 \lambda x \right )-1\right )^{\frac {3}{2}} \sqrt {\cosh \left (2 \lambda x \right )+1}}d x \right )}\right )\]

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6.2.8.6 [585] problem number 6

problem number 585

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["Global`*"]; 
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]];
 

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

Maple ✓

restart; 
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'));
 

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

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6.2.8.7 [586] problem number 7

problem number 586

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["Global`*"]; 
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]];
 

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

Maple ✓

restart; 
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'));
 

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

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