6.2.9 4.2

6.2.9.1 [587] problem number 1

problem number 587

Added January 10, 2019.

Problem 2.4.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.2.9.2 [588] problem number 2

problem number 588

Added January 10, 2019.

Problem 2.4.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.2.9.3 [589] problem number 3

problem number 589

Added January 10, 2019.

Problem 2.4.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + ((a*Cosh[lambda*x]^2 - lambda)*y^2 - a*Cosh[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 *cosh(lambda*x)^2-lambda)*y^2 - a*cosh(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'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( -8\,{ \left ( -1/8\, \left ( \cosh \left ( 2\,\lambda \,x \right ) +1 \right ) \left ( a\cosh \left ( 2\,\lambda \,x \right ) +a-2\,\lambda \right ) \sinh \left ( 2\,\lambda \,x \right ) +y \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{4} \left ( a \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}-\lambda \right ) \right ) \sqrt {-1+\cosh \left ( 2\,\lambda \,x \right ) } \left ( 8\, \left ( -1/8\, \left ( \cosh \left ( 2\,\lambda \,x \right ) +1 \right ) \left ( a\cosh \left ( 2\,\lambda \,x \right ) +a-2\,\lambda \right ) \sinh \left ( 2\,\lambda \,x \right ) +y \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{4} \left ( a \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}-\lambda \right ) \right ) \sqrt {-1+\cosh \left ( 2\,\lambda \,x \right ) }\int \!2\,{\frac { \left ( a\cosh \left ( 2\,\lambda \,x \right ) +a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{ \left ( \cosh \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}\sqrt {-1+\cosh \left ( 2\,\lambda \,x \right ) }}{{\rm e}^{1/2\,{\frac {a\cosh \left ( 2\,\lambda \,x \right ) }{\lambda }}}}}\,{\rm d}x+4\,\sqrt {\cosh \left ( 2\,\lambda \,x \right ) +1}{{\rm e}^{1/2\,{\frac {a\cosh \left ( 2\,\lambda \,x \right ) }{\lambda }}}}\lambda \,\sinh \left ( 2\,\lambda \,x \right ) \left ( a\cosh \left ( 2\,\lambda \,x \right ) +a-2\,\lambda \right ) \right ) ^{-1}} \right ) \]

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6.2.9.4 [590] problem number 4

problem number 590

Added January 10, 2019.

Problem 2.4.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

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

Failed

Maple ✓

restart; 
pde := 2*diff(w(x,y),x)+ ( (a - lambda + a*cosh(lambda*x))*y^2 + a+ lambda- a *cosh(lambda*x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\sqrt {\cosh \left ( \lambda \,x \right ) -1} \left ( \cosh \left ( \lambda \,x \right ) +1 \right ) ^{3/2} \left ( y\cosh \left ( \lambda \,x \right ) +y-\sinh \left ( \lambda \,x \right ) \right ) \left ( \sqrt {\cosh \left ( \lambda \,x \right ) -1} \left ( \left ( \cosh \left ( \lambda \,x \right ) +1 \right ) ^{3/2}\sinh \left ( \lambda \,x \right ) - \left ( \cosh \left ( \lambda \,x \right ) +1 \right ) ^{5/2}y \right ) \int \!{\frac { \left ( a-\lambda +a\cosh \left ( \lambda \,x \right ) \right ) \lambda \,\sinh \left ( \lambda \,x \right ) }{\sqrt {\cosh \left ( \lambda \,x \right ) -1} \left ( \cosh \left ( \lambda \,x \right ) +1 \right ) ^{3/2}}{{\rm e}^{{\frac {a\cosh \left ( \lambda \,x \right ) }{\lambda }}}}}\,{\rm d}x-2\,\sinh \left ( \lambda \,x \right ) {{\rm e}^{{\frac {a\cosh \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \cosh \left ( \lambda \,x \right ) +1 \right ) \lambda \right ) ^{-1}} \right ) \]

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6.2.9.5 [591] problem number 5

problem number 591

Added January 10, 2019.

Problem 2.4.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n + b*x*Cosh[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*cosh(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'));
 

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

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6.2.9.6 [592] problem number 6

problem number 592

Added January 10, 2019.

Problem 2.4.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n + b*x*Cosh[y]^m)*D[w[x, y], x] + Cosh[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*cosh(y)^m)*diff(w(x,y),x)+cosh(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'));
 

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

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6.2.9.7 [593] problem number 7

problem number 593

Added January 10, 2019.

Problem 2.4.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n*y^m + b*x)*D[w[x, y], x] + Cosh[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)+cosh(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'));
 

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

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6.2.9.8 [594] problem number 8

problem number 594

Added January 10, 2019.

Problem 2.4.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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