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'));
 

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

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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'));
 

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

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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]];
 

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

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right ) \lambda -\left (\left (a +\lambda \right ) \tanh \left (\lambda x \right )+y \right ) \operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )}{-\lambda \operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )+\operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right ) \left (\left (a +\lambda \right ) \tanh \left (\lambda x \right )+y \right )}\right )\]

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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 ) = f_{1} \left (\frac {\left (i \lambda -\sqrt {a^{2}+4 a \lambda -\lambda ^{2}}\right ) \left (\frac {\tanh \left (\lambda x \right )}{2}+\frac {1}{2}\right )^{-\frac {\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}}{\lambda }} \left (a +\lambda +\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}\right ) \left (-\left (a -\lambda +y \right ) \left (\sinh \left (\lambda x \right )+\cosh \left (\lambda x \right )\right ) a \left (\lambda +\frac {\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}}{2}\right ) \operatorname {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 {-\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}+2 \lambda }{\lambda }\right ], \frac {\tanh \left (\lambda x \right )}{2}+\frac {1}{2}\right )+\operatorname {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 {\tanh \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cosh \left (\lambda x \right ) \left (i \left (a^{2}+\left (3 \lambda -y \right ) a -2 \lambda \left (y +\lambda \right )\right ) \sqrt {a^{2}+4 a \lambda -\lambda ^{2}}-a^{3}+\left (-y -6 \lambda \right ) a^{2}+\left (-7 \lambda ^{2}-3 \lambda y \right ) a +2 \lambda ^{2} \left (y +\lambda \right )\right )\right )}{\left (a +\lambda -\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}\right ) \left (\sqrt {a^{2}+4 a \lambda -\lambda ^{2}}+i \lambda \right ) \left (\cosh \left (\lambda x \right ) \operatorname {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 {\tanh \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \left (i \left (a^{2}+\left (3 \lambda -y \right ) a -2 \lambda \left (y +\lambda \right )\right ) \sqrt {a^{2}+4 a \lambda -\lambda ^{2}}+a^{3}+\left (y +6 \lambda \right ) a^{2}+\left (7 \lambda ^{2}+3 \lambda y \right ) a -2 \lambda ^{2} \left (y +\lambda \right )\right )+\left (a -\lambda +y \right ) \left (\sinh \left (\lambda x \right )+\cosh \left (\lambda x \right )\right ) \operatorname {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 {\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}+2 \lambda }{\lambda }\right ], \frac {\tanh \left (\lambda x \right )}{2}+\frac {1}{2}\right ) a \left (\lambda -\frac {\sqrt {-a^{2}-4 a \lambda +\lambda ^{2}}}{2}\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'));
 

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

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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'));
 

time expired

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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'));
 

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

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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'));
 

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

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