6.2.12 4.5

6.2.12.1 [607] problem number 1

problem number 607

Added January 10, 2019.

Problem 2.4.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {4 \tan ^{-1}\left (\tanh \left (\frac {\mu y}{2}\right )\right )}{\mu }-\frac {2 a \cosh (\lambda x)}{\lambda }\right )\right \}\right \}\]

Maple ✓

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

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-\cosh \left ( \lambda \,x \right ) a\mu +2\,\arctan \left ( {{\rm e}^{\mu \,y}} \right ) \lambda }{\lambda \,a\mu }} \right ) \]

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6.2.12.2 [608] problem number 2

problem number 608

Added January 10, 2019.

Problem 2.4.5.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {\log \left (\tanh ^2\left (\frac {\mu y}{2}\right )\right )}{\mu }-\frac {2 a \sinh (\lambda x)}{\lambda }\right )\right \}\right \}\]

Maple ✓

restart; 
pde := diff(w(x,y),x)+a*cosh(lambda*x)*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'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac {{{\rm e}^{\lambda \,x}}a\mu -a\mu \,{{\rm e}^{-\lambda \,x}}+4\,\arctanh \left ( {{\rm e}^{\mu \,y}} \right ) \lambda }{\lambda \,a\mu }} \right ) \]

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6.2.12.3 [609] problem number 3

problem number 609

Added January 10, 2019.

Problem 2.4.5.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{-4 \lambda x} \left (16 \lambda ^2 x e^{4 \lambda x} \left (e^{4 \lambda x}+1\right )+y \left (e^{4 \lambda x}+1\right ) \left (e^{4 \lambda x}-1\right )^2+2 \lambda \left (e^{4 \lambda x}-1\right ) \left (-2 e^{4 \lambda x} (2 x y+3)+e^{8 \lambda x}+1\right )\right )}{2 \left (-y e^{4 \lambda x}+2 \lambda \left (e^{4 \lambda x}+1\right )+y\right )}\right )\right \}\right \}\]

Maple ✓

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

\[w \left ( x,y \right ) ={\it \_F1} \left ( {(-4\, \left ( {\rm coth} \left (\lambda \,x\right ) \right ) ^{2}\sinh \left ( \lambda \,x \right ) \lambda + \left ( -4\,y\sinh \left ( \lambda \,x \right ) +8\,\cosh \left ( \lambda \,x \right ) \lambda \right ) {\rm coth} \left (\lambda \,x\right )+4\,\lambda \,\sinh \left ( \lambda \,x \right ) ) \left ( \left ( 4\, \left ( {\rm coth} \left (\lambda \,x\right ) \right ) ^{2}\sinh \left ( \lambda \,x \right ) \lambda + \left ( 4\,y\sinh \left ( \lambda \,x \right ) -8\,\cosh \left ( \lambda \,x \right ) \lambda \right ) {\rm coth} \left (\lambda \,x\right )-4\,\lambda \,\sinh \left ( \lambda \,x \right ) \right ) \ln \left ( {\frac {\cosh \left ( \lambda \,x \right ) -\sinh \left ( \lambda \,x \right ) }{\sinh \left ( \lambda \,x \right ) }} \right ) + \left ( -4\, \left ( {\rm coth} \left (\lambda \,x\right ) \right ) ^{2}\sinh \left ( \lambda \,x \right ) \lambda + \left ( -4\,y\sinh \left ( \lambda \,x \right ) +8\,\cosh \left ( \lambda \,x \right ) \lambda \right ) {\rm coth} \left (\lambda \,x\right )+4\,\lambda \,\sinh \left ( \lambda \,x \right ) \right ) \ln \left ( {\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\sinh \left ( \lambda \,x \right ) }} \right ) + \left ( -\lambda \, \left ( {\rm coth} \left (\lambda \,x\right ) \right ) ^{2}-{\rm coth} \left (\lambda \,x\right )y+\lambda \right ) \cosh \left ( 3\,\lambda \,x \right ) + \left ( \lambda \, \left ( {\rm coth} \left (\lambda \,x\right ) \right ) ^{2}+{\rm coth} \left (\lambda \,x\right )y-\lambda \right ) \cosh \left ( 5\,\lambda \,x \right ) +2\,\lambda \,{\rm coth} \left (\lambda \,x\right ) \left ( \sinh \left ( 5\,\lambda \,x \right ) -4\,\sinh \left ( \lambda \,x \right ) -3\,\sinh \left ( 3\,\lambda \,x \right ) \right ) \right ) ^{-1}} \right ) \]

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6.2.12.4 [610] problem number 4

problem number 610

Added January 10, 2019.

Problem 2.4.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) ={\it \_F1} \left ( { \left ( 2\,a+3\,\lambda \right ) \left ( a \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}+ \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}b-y\cosh \left ( \lambda \,x \right ) \sinh \left ( \lambda \,x \right ) -a \right ) \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2} \left ( {\frac {\cosh \left ( \lambda \,x \right ) }{\sinh \left ( \lambda \,x \right ) }} \right ) ^{{\frac {-2\,a-\lambda }{\lambda }}} \left ( - \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{-2} \right ) ^{{\frac {a+b}{\lambda }}} \left ( 4\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}\lambda \, \left ( b-\lambda /2 \right ) \hypergeom \left ( [2,-1/2\,{\frac {2\,b-3\,\lambda }{\lambda }}],[1/2\,{\frac {2\,a+5\,\lambda }{\lambda }}],{\frac { \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}} \right ) +2\, \left ( \cosh \left ( \lambda \,x \right ) -1 \right ) \left ( \left ( a+b \right ) \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}+y\cosh \left ( \lambda \,x \right ) \sinh \left ( \lambda \,x \right ) -a-\lambda \right ) \left ( \cosh \left ( \lambda \,x \right ) +1 \right ) \left ( 3/2\,\lambda +a \right ) \hypergeom \left ( [1,1/2\,{\frac {-2\,b+\lambda }{\lambda }}],[1/2\,{\frac {2\,a+3\,\lambda }{\lambda }}],{\frac { \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}} \right ) \right ) ^{-1}} \right ) \]

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6.2.12.5 [611] problem number 5

problem number 611

Added January 10, 2019.

Problem 2.4.5.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-\sinh \left ( \beta \,x \right ) a\lambda +\cosh \left ( y\lambda \right ) \beta }{a\beta \,\lambda }} \right ) \]

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6.2.12.6 [612] problem number 6

problem number 612

Added January 10, 2019.

Problem 2.4.5.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n*Cosh[lambda*y]^m + b*x)*D[w[x, y], x] + Sinh[beta*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*cosh(lambda*y)^m+b*x)*diff(w(x,y),x)+sinh(beta*y)^k*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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