6.2.8 4.1

6.2.8.1 [581] problem number 1

problem number 581

Added January 10, 2019.

Problem 2.4.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.2.8.2 [582] problem number 2

problem number 582

Added January 10, 2019.

Problem 2.4.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.2.8.3 [583] problem number 3

problem number 583

Added January 10, 2019.

Problem 2.4.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {2 \lambda e^{\frac {a e^{-\lambda x} \left (e^{2 \lambda x}-1\right )}{\lambda }+\lambda x}}{a e^{2 \lambda x}+a-2 y e^{\lambda x}}-\int _1^{e^{\lambda x}}\frac {e^{\frac {a \left (K[1]^2-1\right )}{\lambda K[1]}}}{K[1]}dK[1]\right )\right \}\right \}\]

Maple ✓

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

\[w \left ( x,y \right ) ={\it \_F1} \left ( -2\,{\sqrt {\sinh \left ( \lambda \,x \right ) +i} \left ( \left ( i \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}-i+2\,\sinh \left ( \lambda \,x \right ) \right ) \left ( a\cosh \left ( \lambda \,x \right ) +y \right ) \HeunC \left ( {\frac {4\,ia}{\lambda }},-1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) -\lambda \,\HeunCPrime \left ( {\frac {4\,ia}{\lambda }},-1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) \left ( i\sinh \left ( \lambda \,x \right ) -1/2\, \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}+1/2 \right ) \cosh \left ( \lambda \,x \right ) \right ) \left ( \left ( \left ( 2\,i \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{3}a+ \left ( i\lambda +2\,a \right ) \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}+ \left ( 2\,ia+2\,\lambda \right ) \sinh \left ( \lambda \,x \right ) -i\lambda +2\,a \right ) \cosh \left ( \lambda \,x \right ) +2\, \left ( 1+i\sinh \left ( \lambda \,x \right ) \right ) y \left ( \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}+1 \right ) \right ) \HeunC \left ( {\frac {4\,ia}{\lambda }},1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) -\lambda \, \left ( -\sinh \left ( \lambda \,x \right ) +i \right ) \left ( \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}+1 \right ) \HeunCPrime \left ( {\frac {4\,ia}{\lambda }},1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) \cosh \left ( \lambda \,x \right ) \right ) ^{-1}} \right ) \]

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6.2.8.4 [584] problem number 4

problem number 584

Added January 10, 2019.

Problem 2.4.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + lambda*(Sinh[lambda*x]*y^2 - Sinh[lambda*x]^3)*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)+ lambda*(sinh(lambda*x)*y^2 - sinh(lambda*x)^3)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( y-\cosh \left ( \lambda \,x \right ) \right ) \sqrt {\pi }}{\sqrt {\pi }\cosh \left ( \lambda \,x \right ) \erfi \left ( \cosh \left ( \lambda \,x \right ) \right ) -\sqrt {\pi }y\erfi \left ( \cosh \left ( \lambda \,x \right ) \right ) -2\,{{\rm e}^{ \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}}}} \right ) \]

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6.2.8.5 [585] problem number 5

problem number 585

Added January 10, 2019.

Problem 2.4.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + ((a*Sinh[lambda*x]^2 - lambda)*y^2 - a*Sinh[lambda*x]^2 + lambda - a)*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)+ ((a*sinh(lambda*x)^2-lambda)*y^2 - a*sinh(lambda*x)^2 + lambda - a)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( -2\,{ \left ( -1+\cosh \left ( 2\,\lambda \,x \right ) \right ) \left ( \left ( -1/2\,a\sinh \left ( 2\,\lambda \,x \right ) +y \left ( \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}a-\lambda \right ) \right ) \cosh \left ( 2\,\lambda \,x \right ) + \left ( a/2+\lambda \right ) \sinh \left ( 2\,\lambda \,x \right ) -y \left ( \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}a-\lambda \right ) \right ) \sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1} \left ( 2\, \left ( -1+\cosh \left ( 2\,\lambda \,x \right ) \right ) \left ( \left ( -1/2\,a\sinh \left ( 2\,\lambda \,x \right ) +y \left ( \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}a-\lambda \right ) \right ) \cosh \left ( 2\,\lambda \,x \right ) + \left ( a/2+\lambda \right ) \sinh \left ( 2\,\lambda \,x \right ) -y \left ( \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}a-\lambda \right ) \right ) \sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}\int \!2\,{\frac { \left ( \cosh \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cosh \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x+4\,{{\rm e}^{1/2\,{\frac {\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}\lambda \,\sinh \left ( 2\,\lambda \,x \right ) \sqrt {-1+\cosh \left ( 2\,\lambda \,x \right ) } \left ( \cosh \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \right ) ^{-1}} \right ) \]

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6.2.8.6 [586] problem number 6

problem number 586

Added January 10, 2019.

Problem 2.4.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.2.8.7 [587] problem number 7

problem number 587

Added January 10, 2019.

Problem 2.4.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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