6.6.9 4.3

6.6.9.1 [1476] Problem 1

problem number 1476

Added May 19, 2019.

Problem Chapter 6.4.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y , z\right ) = f_{1} \left (\frac {a y -b x}{a}, \frac {-x c \gamma +\ln \left (-\frac {\tanh \left (\gamma z \right )}{\sqrt {-\operatorname {sech}\left (\gamma z \right )^{2}}}\right ) a}{c \gamma }\right )\]

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6.6.9.2 [1477] Problem 2

problem number 1477

Added May 19, 2019.

Problem Chapter 6.4.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*tanh(beta*x)*diff(w(x,y,z),y)+c*tanh(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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {y a \beta -b \ln \left (\cosh \left (\beta x \right )\right )}{a \beta }, \frac {z a \lambda -c \ln \left (\cosh \left (\lambda x \right )\right )}{a \lambda }\right )\]

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6.6.9.3 [1478] Problem 3

problem number 1478

Added May 19, 2019.

Problem Chapter 6.4.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*tanh(beta*y)*diff(w(x,y,z),y)+c*tanh(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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-x b \beta +\ln \left (-\frac {\tanh \left (\beta y \right )}{\sqrt {-\operatorname {sech}\left (\beta y \right )^{2}}}\right ) a}{b \beta }, \frac {z a \lambda -c \ln \left (\cosh \left (\lambda x \right )\right )}{a \lambda }\right )\]

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6.6.9.4 [1479] Problem 4

problem number 1479

Added May 19, 2019.

Problem Chapter 6.4.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-x b \beta +\ln \left (-\frac {\tanh \left (\beta y \right )}{\sqrt {-\operatorname {sech}\left (\beta y \right )^{2}}}\right ) a}{b \beta }, \frac {-x c \gamma +\ln \left (-\frac {\tanh \left (\gamma z \right )}{\sqrt {-\operatorname {sech}\left (\gamma z \right )^{2}}}\right ) a}{c \gamma }\right )\]

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6.6.9.5 [1480] Problem 5

problem number 1480

Added May 19, 2019.

Problem Chapter 6.4.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*Tanh[lambda*x]*D[w[x, y,z], x] + b*Tanh[beta*y]*D[w[x, y,z], y] +c*Tanh[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*tanh(lambda*x)*diff(w(x,y,z),x)+ b*tanh(beta*y)*diff(w(x,y,z),y)+c*tanh(gamma*z)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-b \beta \ln \left ({\mathrm e}^{2 \lambda x}-1\right )+\ln \left (i \sinh \left (\beta y \right )\right ) a \lambda +b \beta \left (\lambda x +\ln \left (2\right )\right )}{\lambda b \beta }, \frac {-c \gamma \ln \left ({\mathrm e}^{2 \lambda x}-1\right )+\ln \left (i \sinh \left (\gamma z \right )\right ) a \lambda +c \gamma \left (\lambda x +\ln \left (2\right )\right )}{\lambda c \gamma }\right )\]

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6.6.9.6 [1481] Problem 6

problem number 1481

Added May 19, 2019.

Problem Chapter 6.4.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*Tanh[beta*y]*D[w[x, y,z], x] + b*Tanh[lambda*x]*D[w[x, y,z], y] +c*Tanh[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*tanh(beta*y)*diff(w(x,y,z),x)+ b*tanh(lambda*x)*diff(w(x,y,z),y)+c*tanh(gamma*z)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-b \beta \ln \left ({\mathrm e}^{2 \lambda x}+1\right )+\ln \left (\cosh \left (\beta y \right )\right ) a \lambda +b \beta \left (\lambda x +\ln \left (2\right )\right )}{\lambda b \beta }, \frac {\ln \left (\frac {\tanh \left (\gamma z \right )}{\sqrt {-{\mathrm e}^{\frac {2 \int _{}^{x}\frac {1}{\sqrt {1-\operatorname {sech}\left (\beta y \right )^{2} {\mathrm e}^{\frac {2 b \beta \left (-x +\textit {\_a} \right )}{a}} \left ({\mathrm e}^{2 \lambda x}+1\right )^{\frac {2 b \beta }{a \lambda }} \left ({\mathrm e}^{2 \lambda \textit {\_a}}+1\right )^{-\frac {2 b \beta }{a \lambda }}}}d \textit {\_a} c \gamma }{a}} \operatorname {sech}\left (\gamma z \right )^{2}}}\right ) a}{c \gamma }\right )\]

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