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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {y \lambda -a \sinh \left (\lambda x \right )}{\lambda }\right )\]

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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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {a \lambda x+\cot ^{-1}(\sinh (\lambda y))}{\lambda }\right )\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-x a \lambda +2 \arctan \left ({\mathrm e}^{\lambda y}\right )}{a \lambda }\right )\]

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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]];
 

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

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'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {2 \cosh \left (\lambda x \right )^{2} y -\sinh \left (2 \lambda x \right )}{2 \lambda \left (-2 \int -{\mathrm e}^{\frac {\cosh \left (2 \lambda x \right ) a}{2 \lambda }} \left (a -\lambda \operatorname {sech}\left (\lambda x \right )^{2}\right )d x \cosh \left (\lambda x \right )^{2} y +\sinh \left (2 \lambda x \right ) \int -{\mathrm e}^{\frac {\cosh \left (2 \lambda x \right ) a}{2 \lambda }} \left (a -\lambda \operatorname {sech}\left (\lambda x \right )^{2}\right )d x +2 \,{\mathrm e}^{\frac {\cosh \left (2 \lambda x \right ) a}{2 \lambda }}\right )}\right )\]

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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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {2 \cosh \left (\frac {\lambda x}{2}\right ) \left (y \cosh \left (\frac {\lambda x}{2}\right )-\sinh \left (\frac {\lambda x}{2}\right )\right )}{(y \cosh (\lambda x)-\sinh (\lambda x)+y) \int _1^x-e^{\frac {2 a \cosh ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} (\cosh (\lambda K[1]) a+a-\lambda ) \text {sech}^2\left (\frac {1}{2} \lambda K[1]\right )dK[1]-4 e^{\frac {2 a \cosh ^2\left (\frac {\lambda x}{2}\right )}{\lambda }}}\right )\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {2 \cosh \left (\frac {\lambda x}{2}\right ) \left (\cosh \left (\frac {\lambda x}{2}\right ) y -\sinh \left (\frac {\lambda x}{2}\right )\right )}{\lambda \left (\cosh \left (\frac {\lambda x}{2}\right ) \left (\cosh \left (\frac {\lambda x}{2}\right ) y -\sinh \left (\frac {\lambda x}{2}\right )\right ) \int {\mathrm e}^{\frac {\cosh \left (\lambda x \right ) a}{\lambda }} \left (-2 a +\lambda \operatorname {sech}\left (\frac {\lambda x}{2}\right )^{2}\right )d x -2 \,{\mathrm e}^{\frac {\cosh \left (\lambda x \right ) a}{\lambda }}\right )}\right )\]

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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'));
 

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

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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'));
 

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

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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'));
 

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

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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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-\sinh \left (\lambda x \right ) a \mu +\sinh \left (\mu y \right ) \lambda }{\lambda a \mu }\right )\]

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