6.2.10 4.3

6.2.10.1 [595] problem number 1

problem number 595

Added January 10, 2019.

Problem 2.4.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + a \tanh (\lambda x) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + a*Tanh[lambda*x]*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

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

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {a \ln \left (\tanh \left (\lambda x \right )-1\right )+a \ln \left (\tanh \left (\lambda x \right )+1\right )+2 \lambda y}{2 \lambda }\right )\]

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6.2.10.2 [596] problem number 2

problem number 596

Added January 10, 2019.

Problem 2.4.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + a \tanh (\lambda y) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + a*Tanh[lambda*y]*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

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

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {a \ln \left (\tanh \left (\lambda x \right )-1\right )+a \ln \left (\tanh \left (\lambda x \right )+1\right )+2 \lambda y}{2 \lambda }\right )\]

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6.2.10.3 [597] problem number 3

problem number 597

Added January 10, 2019.

Problem 2.4.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( y^2+a \lambda - a (a+\lambda ) \tanh ^2(\lambda x) \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + a*lambda - a*(a + lambda)*Tanh[lambda*x]^2)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {\lambda e^{-2 a x} \left (\, _2F_1\left (-\frac {2 a}{\lambda },-\frac {a}{\lambda };1-\frac {a}{\lambda };-e^{2 \lambda x}\right ) \left (a \left (e^{2 \lambda x}-1\right )-y \left (e^{2 \lambda x}+1\right )\right )+2 a \left (e^{2 \lambda x}+1\right )^{\frac {2 a}{\lambda }+1}\right )}{2 a \left (a \left (-e^{2 \lambda x}\right )+a+y e^{2 \lambda x}+y\right )}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := diff(w(x,y),x)+( y^2+a*lambda - a*(a+lambda)*tanh(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'));
 

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

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6.2.10.4 [598] problem number 4

problem number 598

Added January 10, 2019.

Problem 2.4.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( y^2+3 a \lambda - \lambda ^2 -a(a+\lambda ) \tanh ^2(\lambda x) \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + 3*a*lambda - lambda^2 - a*(a + lambda)*Tanh[lambda*x])*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)+( y^2+3*a*lambda - lambda^2 -a*(a+lambda)*tanh(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 {\left (\lambda +\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}\right ) \left (-\left (a -\lambda +y \right ) \left (\cosh \left (\lambda x \right )+\sinh \left (\lambda x \right )\right ) \left (\lambda +\frac {\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}}{2}\right ) a \hypergeom \left (\left [\frac {a +\lambda -\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}}{2 \lambda }, \frac {a +\lambda -\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}}{2 \lambda }\right ], \left [\frac {2 \lambda -\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}}{\lambda }\right ], \frac {\cosh \left (\lambda x \right )+\sinh \left (\lambda x \right )}{2 \cosh \left (\lambda x \right )}\right )+\left (-a^{3}+\left (-6 \lambda -y \right ) a^{2}+2 \left (\lambda +y \right ) \lambda ^{2}+\left (-7 \lambda ^{2}-3 \lambda y \right ) a +\left (i a^{2}+\left (3 i \lambda -i y \right ) a -2 i \left (\lambda +y \right ) \lambda \right ) \sqrt {a^{2}+4 a \lambda -\lambda ^{2}}\right ) \hypergeom \left (\left [-\frac {-a +\lambda +\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}}{2 \lambda }, -\frac {-a +\lambda +\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}}{2 \lambda }\right ], \left [\frac {\lambda -\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}}{\lambda }\right ], \frac {\cosh \left (\lambda x \right )+\sinh \left (\lambda x \right )}{2 \cosh \left (\lambda x \right )}\right ) \cosh \left (\lambda x \right )\right ) \left (a +\lambda +\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}\right ) 2^{-\frac {\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}}{\lambda }} \left (\frac {\cosh \left (\lambda x \right )+\sinh \left (\lambda x \right )}{\cosh \left (\lambda x \right )}\right )^{-\frac {\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}}{\lambda }}}{\left (a +\lambda -\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}\right ) \left (\lambda -\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}\right ) \left (\left (a -\lambda +y \right ) \left (\lambda -\frac {\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}}{2}\right ) \left (\cosh \left (\lambda x \right )+\sinh \left (\lambda x \right )\right ) a \hypergeom \left (\left [\frac {a +\lambda +\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}}{2 \lambda }, \frac {a +\lambda +\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}}{2 \lambda }\right ], \left [\frac {2 \lambda +\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}}{\lambda }\right ], \frac {\cosh \left (\lambda x \right )+\sinh \left (\lambda x \right )}{2 \cosh \left (\lambda x \right )}\right )+\left (a^{3}+\left (6 \lambda +y \right ) a^{2}-2 \left (\lambda +y \right ) \lambda ^{2}+\left (7 \lambda ^{2}+3 \lambda y \right ) a +\left (i a^{2}+\left (3 i \lambda -i y \right ) a -2 i \left (\lambda +y \right ) \lambda \right ) \sqrt {a^{2}+4 a \lambda -\lambda ^{2}}\right ) \hypergeom \left (\left [\frac {a -\lambda +\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}}{2 \lambda }, \frac {a -\lambda +\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}}{2 \lambda }\right ], \left [\frac {\lambda +\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}}{\lambda }\right ], \frac {\cosh \left (\lambda x \right )+\sinh \left (\lambda x \right )}{2 \cosh \left (\lambda x \right )}\right ) \cosh \left (\lambda x \right )\right )}\right )\]

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6.2.10.5 [599] problem number 5

problem number 599

Added January 10, 2019.

Problem 2.4.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ \left ( a x^n + b x \tanh ^m(y)\right ) w_x + y^k w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n + b*x*Tanh[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]];
 

Failed

Maple ✓

restart; 
pde := ( a*x^n + b*x*tanh(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'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\left (n -1\right ) a \left (\int y^{-k} {\mathrm e}^{\left (n -1\right ) b \left (\int y^{-k} \left (\tanh ^{m}y \right )d y \right )}d y \right )+x^{-n +1} {\mathrm e}^{\left (n -1\right ) b \left (\int y^{-k} \left (\tanh ^{m}y \right )d y \right )}\right )\]

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6.2.10.6 [600] problem number 6

problem number 600

Added January 10, 2019.

Problem 2.4.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ \left ( a x^n + b x \tanh ^m(y)\right ) w_x + \tanh ^k(\lambda y) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n + b*x*Tanh[y]^m)*D[w[x, y], x] + Tanh[lambda*y]^k*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := ( a*x^n + b*x*tanh(y)^m)*diff(w(x,y),x)+tanh(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'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\left (n -1\right ) a \left (\int \left (\tanh ^{-k}\left (\lambda y \right )\right ) {\mathrm e}^{\left (n -1\right ) b \left (\int \left (\tanh ^{m}y \right ) \left (\tanh ^{-k}\left (\lambda y \right )\right )d y \right )}d y \right )+x^{-n +1} {\mathrm e}^{\left (n -1\right ) b \left (\int \left (\tanh ^{m}y \right ) \left (\tanh ^{-k}\left (\lambda y \right )\right )d y \right )}\right )\]

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6.2.10.7 [601] problem number 7

problem number 601

Added January 10, 2019.

Problem 2.4.3.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ \left ( a x^n y^m + b x\right ) w_x + \tanh ^k(\lambda y) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n*y^m + b*x)*D[w[x, y], x] + Tanh[lambda*y]^k*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := ( a*x^n*y^m + b*x)*diff(w(x,y),x)+tanh(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'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\left (n -1\right ) a \left (\int y^{m} \left (\tanh ^{-k}\left (\lambda y \right )\right ) {\mathrm e}^{\left (n -1\right ) b \left (\int \left (\tanh ^{-k}\left (\lambda y \right )\right )d y \right )}d y \right )+x^{-n +1} {\mathrm e}^{\left (n -1\right ) b \left (\int \left (\tanh ^{-k}\left (\lambda y \right )\right )d y \right )}\right )\]

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6.2.10.8 [602] problem number 8

problem number 602

Added January 10, 2019.

Problem 2.4.3.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ \left ( a x^n \tanh ^m y + b x\right ) w_x + y^k w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  (a*x^n*Tanh[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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\int _1^yK[1]^{-k} \tanh ^m(K[1])dK[1]+\frac {x^{1-n}}{a (n-1)}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := ( a*x^n*tanh(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'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\left (n -1\right ) a \left (\int y^{-k} \left (\tanh ^{m}y \right )d y \right )+x^{-n +1}\right )\]

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