7.6.8 4.2

7.6.8.1 [1472] Problem 1

problem number 1472

Added May 19, 2019.

Problem Chapter 6.4.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a w_x + b w_y + c \cosh (\beta x) w_z = 0 \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {a y -b x}{a}, \frac {a \beta z -c \sinh \left (\beta x \right )}{a \beta }\right )\]

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7.6.8.2 [1473] Problem 2

problem number 1473

Added May 19, 2019.

Problem Chapter 6.4.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a w_x + b \cosh (\beta x) w_y + c \cosh (\lambda x) w_z = 0 \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {a \beta y -b \sinh \left (\beta x \right )}{a \beta }, \frac {a \lambda z -c \sinh \left (\lambda x \right )}{a \lambda }\right )\]

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7.6.8.3 [1474] Problem 3

problem number 1474

Added May 19, 2019.

Problem Chapter 6.4.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a w_x + b \cosh (\beta y) w_y + c \cosh (\lambda x) w_z = 0 \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {-b \beta x +2 a \arctan \left ({\mathrm e}^{\beta y}\right )}{b \beta }, \frac {a \lambda z -c \sinh \left (\lambda x \right )}{a \lambda }\right )\]

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7.6.8.4 [1475] Problem 4

problem number 1475

Added May 19, 2019.

Problem Chapter 6.4.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a w_x + b \cosh (\beta y) w_y + c \cosh (\gamma z) w_z = 0 \]

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {2 \tan ^{-1}\left (\tanh \left (\frac {\beta y}{2}\right )\right )}{\beta }-\frac {b x}{a},\frac {2 \tan ^{-1}\left (\tanh \left (\frac {\gamma z}{2}\right )\right )}{\gamma }-\frac {c x}{a}\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {-b \beta x +2 a \arctan \left ({\mathrm e}^{\beta y}\right )}{b \beta }, \frac {2 a \arctan \left ({\mathrm e}^{\gamma z}\right )-\gamma c x}{\gamma c}\right )\]

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7.6.8.5 [1476] Problem 5

problem number 1476

Added May 19, 2019.

Problem Chapter 6.4.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a \cosh (\lambda x) w_x + b \cosh (\beta y) w_y + c \cosh (\gamma z) w_z = 0 \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {2 a \lambda \arctan \left ({\mathrm e}^{\beta y}\right )-2 b \beta \arctan \left ({\mathrm e}^{\lambda x}\right )}{b \beta \lambda }, \frac {2 a \lambda \arctan \left ({\mathrm e}^{\gamma z}\right )-2 \gamma c \arctan \left ({\mathrm e}^{\lambda x}\right )}{\gamma c \lambda }\right )\]

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7.6.8.6 [1477] Problem 6

problem number 1477

Added May 19, 2019.

Problem Chapter 6.4.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a \cosh (\beta y) w_x + b \cosh (\lambda x) w_y + c \cosh (\gamma z) w_z = 0 \]

Mathematica ✗

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

Failed

Maple ✓

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

\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {a \lambda \sinh \left (\beta y \right )-b \beta \sinh \left (\lambda x \right )}{b \beta \lambda }, -\frac {4 \left (-\frac {\left (\left (a \lambda -b \beta \right ) \sqrt {-\left (\sinh ^{2}\left (\beta y \right )\right )-1}+\sqrt {\sinh ^{2}\left (\beta y \right )+1}\, \left (a \lambda \sinh \left (\beta y \right )-b \beta \sinh \left (\lambda x \right )\right )\right ) \sqrt {\left (-\sinh \left (\lambda x \right )+i\right ) \left (\sinh \left (\lambda x \right )+i\right ) \left (-\sinh \left (\beta y \right )+i\right ) \left (\sinh \left (\beta y \right )+i\right )}\, a \lambda \arctan \left ({\mathrm e}^{\gamma z}\right ) \cosh \left (\lambda x \right )}{2}+\sqrt {-\frac {\left (a \lambda \sinh \left (\beta y \right )-b \beta \sinh \left (\lambda x \right )+i a \lambda -i b \beta \right ) \left (-\sinh \left (\lambda x \right )+i\right )}{\left (a \lambda \sinh \left (\beta y \right )-b \beta \sinh \left (\lambda x \right )+i a \lambda +i b \beta \right ) \left (\sinh \left (\lambda x \right )+i\right )}}\, \sqrt {\frac {i \left (\sinh \left (\beta y \right )+i\right ) a \lambda }{\left (a \lambda \sinh \left (\beta y \right )-b \beta \sinh \left (\lambda x \right )+i a \lambda +i b \beta \right ) \left (\sinh \left (\lambda x \right )+i\right )}}\, \sqrt {\left (\sinh ^{2}\left (\beta y \right )+1\right ) \left (\cosh ^{2}\left (\lambda x \right )\right )}\, \left (\left (-\frac {a \lambda \sinh \left (\beta y \right ) \left (\sinh ^{2}\left (\lambda x \right )\right )}{2}+\frac {b \beta \left (\sinh ^{3}\left (\lambda x \right )\right )}{2}+\frac {a \lambda \sinh \left (\beta y \right )}{2}+\frac {b \beta \sinh \left (\lambda x \right )}{2}\right ) \sqrt {-\frac {i \left (-\sinh \left (\beta y \right )+i\right ) a \lambda }{\left (-a \lambda \sinh \left (\beta y \right )+b \beta \sinh \left (\lambda x \right )+i a \lambda -i b \beta \right ) \left (\sinh \left (\lambda x \right )+i\right )}}+\left (\left (\sinh \left (\beta y \right )+i\right ) a \lambda \sinh \left (\lambda x \right )-\frac {a \lambda }{2}-\frac {b \beta }{2}+\left (\frac {a \lambda }{2}-\frac {b \beta }{2}\right ) \left (\sinh ^{2}\left (\lambda x \right )\right )\right ) \sqrt {\frac {i \left (-\sinh \left (\beta y \right )+i\right ) a \lambda }{\left (-a \lambda \sinh \left (\beta y \right )+b \beta \sinh \left (\lambda x \right )+i a \lambda -i b \beta \right ) \left (\sinh \left (\lambda x \right )+i\right )}}\right ) \gamma c \EllipticF \left (\sqrt {-\frac {\left (a \lambda \sinh \left (\beta y \right )-b \beta \sinh \left (\lambda x \right )+i a \lambda -i b \beta \right ) \left (-\sinh \left (\lambda x \right )+i\right )}{\left (a \lambda \sinh \left (\beta y \right )-b \beta \sinh \left (\lambda x \right )+i a \lambda +i b \beta \right ) \left (\sinh \left (\lambda x \right )+i\right )}}, \sqrt {-\frac {a^{2} \lambda ^{2}+2 a b \beta \lambda +b^{2} \beta ^{2}+\left (a \lambda \sinh \left (\beta y \right )-b \beta \sinh \left (\lambda x \right )\right )^{2}}{\left (-a \lambda \sinh \left (\beta y \right )+b \beta \sinh \left (\lambda x \right )+i a \lambda -i b \beta \right ) \left (a \lambda \sinh \left (\beta y \right )-b \beta \sinh \left (\lambda x \right )+i a \lambda -i b \beta \right )}}\right )\right )}{\sqrt {\sinh ^{2}\left (\beta y \right )+1}\, \sqrt {\left (-\sinh \left (\lambda x \right )+i\right ) \left (\sinh \left (\lambda x \right )+i\right ) \left (-\sinh \left (\beta y \right )+i\right ) \left (\sinh \left (\beta y \right )+i\right )}\, \gamma \left (a \lambda \sinh \left (\beta y \right )-b \beta \sinh \left (\lambda x \right )+i a \lambda -i b \beta \right ) c \lambda \cosh \left (\lambda x \right )}\right )\]

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