6.2.11 4.4

6.2.11.1 [603] problem number 1

problem number 603

Added January 10, 2019.

Problem 2.4.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.2.11.2 [604] problem number 2

problem number 604

Added January 10, 2019.

Problem 2.4.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.2.11.3 [605] problem number 3

problem number 605

Added January 10, 2019.

Problem 2.4.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.2.11.4 [606] problem number 4

problem number 606

Added January 10, 2019.

Problem 2.4.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + 3*a*lambda - lambda^2 - a*(a + lambda)*Coth[lambda*x]^2)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {\lambda e^{2 x (\lambda -a)} \left ((a-2 \lambda ) F_1\left (1-\frac {a}{\lambda };-\frac {2 a}{\lambda },2;2-\frac {a}{\lambda };e^{2 \lambda x},-e^{2 \lambda x}\right ) \left (a \left (2 e^{2 \lambda x}+3 e^{4 \lambda x}-1\right )-\left (e^{2 \lambda x}-1\right ) \left (y e^{2 \lambda x}+3 \lambda e^{2 \lambda x}+\lambda +y\right )\right )-4 (\lambda -a) e^{2 \lambda x} \left (e^{4 \lambda x}-1\right ) \left (a F_1\left (2-\frac {a}{\lambda };1-\frac {2 a}{\lambda },2;3-\frac {a}{\lambda };e^{2 \lambda x},-e^{2 \lambda x}\right )+\lambda F_1\left (2-\frac {a}{\lambda };-\frac {2 a}{\lambda },3;3-\frac {a}{\lambda };e^{2 \lambda x},-e^{2 \lambda x}\right )\right )\right )}{2 \left (a^2-3 a \lambda +2 \lambda ^2\right ) \left (a \left (e^{2 \lambda x}+1\right )^2-\left (e^{2 \lambda x}-1\right ) \left (y \left (e^{2 \lambda x}+1\right )+\lambda \left (e^{2 \lambda x}-1\right )\right )\right )}\right )\right \}\right \}\]

Maple ✓

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

\[w \left ( x,y \right ) ={\it \_F1} \left ( { \left ( \left ( -a+\sqrt {{a}^{2}+{\lambda }^{2}}-\lambda \right ) \LegendreP \left ( {\frac {a+\lambda }{\lambda }},{\frac {\sqrt {{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) + \left ( \left ( a+\lambda \right ) {\rm coth} \left (\lambda \,x\right )+y \right ) \LegendreP \left ( {\frac {a}{\lambda }},{\frac {\sqrt {{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) \right ) \left ( \left ( a+\lambda -\sqrt {{a}^{2}+{\lambda }^{2}} \right ) \LegendreQ \left ( {\frac {a+\lambda }{\lambda }},{\frac {\sqrt {{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) -\LegendreQ \left ( {\frac {a}{\lambda }},{\frac {\sqrt {{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) \left ( \left ( a+\lambda \right ) {\rm coth} \left (\lambda \,x\right )+y \right ) \right ) ^{-1}} \right ) \]

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