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'));
 

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

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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'));
 

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

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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'));
 

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

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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 ) ={\it \_F1} \left ( {\frac { \left ( a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}} \right ) \left ( \sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda \right ) }{ \left ( -\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+a+\lambda \right ) \left ( -\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda \right ) } \left ( - \left ( \lambda +1/2\,\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}} \right ) a \left ( a+y-\lambda \right ) \left ( \sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) \right ) \hypergeom \left ( [1/2\,{\frac {-\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+a+\lambda }{\lambda }},1/2\,{\frac {-\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+a+\lambda }{\lambda }}],[{\frac {-\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }}],1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }} \right ) +\hypergeom \left ( [-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }}],[{\frac {-\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }}],1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }} \right ) \cosh \left ( \lambda \,x \right ) \left ( \left ( i{a}^{2}+ \left ( -iy+3\,i\lambda \right ) a-2\,i \left ( \lambda +y \right ) \lambda \right ) \sqrt {{a}^{2}+4\,a\lambda -{\lambda }^{2}}-{a}^{3}+ \left ( -y-6\,\lambda \right ) {a}^{2}+ \left ( -7\,{\lambda }^{2}-3\,y\lambda \right ) a+2\,{\lambda }^{2} \left ( \lambda +y \right ) \right ) \right ) {2}^{-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }}} \left ( {\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }} \right ) ^{-{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }}} \left ( \left ( \left ( i{a}^{2}+ \left ( -iy+3\,i\lambda \right ) a-2\,i \left ( \lambda +y \right ) \lambda \right ) \sqrt {{a}^{2}+4\,a\lambda -{\lambda }^{2}}+{a}^{3}+ \left ( y+6\,\lambda \right ) {a}^{2}+ \left ( 7\,{\lambda }^{2}+3\,y\lambda \right ) a-2\,{\lambda }^{2} \left ( \lambda +y \right ) \right ) \cosh \left ( \lambda \,x \right ) \hypergeom \left ( [1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a-\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }}],[{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }}],1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }} \right ) + \left ( \lambda -1/2\,\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}} \right ) \hypergeom \left ( [1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac {a+\lambda +\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }}],[{\frac {\sqrt {-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }}],1/2\,{\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }} \right ) a \left ( a+y-\lambda \right ) \left ( \sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) \right ) \right ) ^{-1}} \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'));
 

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

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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'));
 

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

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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'));
 

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

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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'));
 

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

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