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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {y \lambda -\cosh \left (\lambda x \right ) a}{\lambda }\right )\]

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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]];
 

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

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-x a \mu -2 \,\operatorname {arctanh}\left ({\mathrm e}^{\mu y}\right )}{a \mu }\right )\]

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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'));
 

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

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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'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {\sqrt {\pi }\, \left (y -\cosh \left (\lambda x \right )\right )}{2 \,{\mathrm e}^{\frac {\cosh \left (2 \lambda x \right )}{2}+\frac {1}{2}}+\sqrt {\pi }\, \left (y -\cosh \left (\lambda x \right )\right ) \operatorname {erfi}\left (\cosh \left (\lambda x \right )\right )}\right )\]

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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]];
 

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

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-2 \sinh \left (\lambda x \right )^{2} y +\sinh \left (2 \lambda x \right )}{2 \lambda \left (-2 \sinh \left (\lambda x \right )^{2} y \int -{\mathrm e}^{\frac {\cosh \left (2 \lambda x \right ) a}{2 \lambda }} \left (a -\lambda \operatorname {csch}\left (\lambda x \right )^{2}\right )d x +\sinh \left (2 \lambda x \right ) \int -{\mathrm e}^{\frac {\cosh \left (2 \lambda x \right ) a}{2 \lambda }} \left (a -\lambda \operatorname {csch}\left (\lambda x \right )^{2}\right )d x +2 \,{\mathrm e}^{\frac {\cosh \left (2 \lambda x \right ) a}{2 \lambda }}\right )}\right )\]

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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]];
 

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

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-\ln \left (\tanh \left (\frac {\lambda x}{2}\right )\right ) a \mu -2 \,\operatorname {arctanh}\left ({\mathrm e}^{\mu y}\right ) \lambda }{\lambda a \mu }\right )\]

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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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-\cosh \left (\lambda x \right ) a \mu +\cosh \left (\mu y \right ) \lambda }{\lambda a \mu }\right )\]

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