7.6.9 4.3

7.6.9.1 [1478] Problem 1

problem number 1478

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'));
 

\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {a y -b x}{a}, \RootOf \left (\gamma z +\arctanh \left (\frac {\sqrt {\left ({\mathrm e}^{\frac {2 \gamma \left (\textit {\_Z} +x \right ) c}{a}}-1\right ) {\mathrm e}^{-\frac {2 \gamma \left (\textit {\_Z} +x \right ) c}{a}}}\, {\mathrm e}^{\frac {2 \gamma \left (\textit {\_Z} +x \right ) c}{a}}}{{\mathrm e}^{\frac {2 \gamma \left (\textit {\_Z} +x \right ) c}{a}}-1}\right )\right )\right )\]

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7.6.9.2 [1479] Problem 2

problem number 1479

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'));
 

\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {2 a \beta y +b \ln \left (\tanh \left (\beta x \right )-1\right )+b \ln \left (\tanh \left (\beta x \right )+1\right )}{2 a \beta }, \frac {2 a \lambda z +c \ln \left (\tanh \left (\lambda x \right )-1\right )+c \ln \left (\tanh \left (\lambda x \right )+1\right )}{2 a \lambda }\right )\]

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7.6.9.3 [1480] Problem 3

problem number 1480

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'));
 

\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {2 b \beta x +a \ln \left (\tanh \left (\beta y \right )-1\right )+a \ln \left (\tanh \left (\beta y \right )+1\right )-2 a \ln \left (\tanh \left (\beta y \right )\right )}{2 b \beta }, \frac {a \lambda z -c \ln \left (\cosh \left (\lambda x \right )\right )}{a \lambda }\right )\]

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7.6.9.4 [1481] Problem 4

problem number 1481

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]];
 

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

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'));
 

\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {2 b \beta x +a \ln \left (\tanh \left (\beta y \right )-1\right )+a \ln \left (\tanh \left (\beta y \right )+1\right )-2 a \ln \left (\tanh \left (\beta y \right )\right )}{2 b \beta }, \frac {b \beta \ln \left (\sqrt {-\left (-\frac {1}{\left ({\mathrm e}^{2 \beta y}-1\right )^{2}}\right )^{\frac {\gamma c}{b \beta }}}\, \sinh \left (\gamma z \right )\right )+\gamma \left (\beta y +\ln \left (2\right )\right ) c}{\gamma \beta c}\right )\]

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7.6.9.5 [1482] Problem 5

problem number 1482

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'));
 

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

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7.6.9.6 [1483] Problem 6

problem number 1483

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'));
 

\[w \left (x , y , z\right ) = c_{1} \left (\tanh \left (\gamma z \right )-1\right )^{-\frac {\textit {\_c}_{3}}{2 \gamma }} \left (\tanh \left (\gamma z \right )+1\right )^{-\frac {\textit {\_c}_{3}}{2 \gamma }} \left (\tanh ^{\frac {\textit {\_c}_{3}}{\gamma }}\left (\gamma z \right )\right ) \textit {\_F5} \left (\frac {a \ln \left (\RootOf \left (\beta y -\mathrm {arccosh}\left (\textit {\_Z} 2^{-\frac {b \beta }{a \lambda }} \left ({\mathrm e}^{2 \lambda x}+1\right )^{\frac {b \beta }{a \lambda }} {\mathrm e}^{-\frac {b \beta x}{a}}\right )\right )\right )}{b \beta }\right ) {\mathrm e}^{-\frac {c \textit {\_c}_{3} 2^{-\frac {b \beta }{a \lambda }} \RootOf \left (\beta y -\mathrm {arccosh}\left (\textit {\_Z} 2^{-\frac {b \beta }{a \lambda }} \left ({\mathrm e}^{2 \lambda x}+1\right )^{\frac {b \beta }{a \lambda }} {\mathrm e}^{-\frac {b \beta x}{a}}\right )\right ) \left (\int _{}^{x}\frac {\left ({\mathrm e}^{2 \textit {\_a} \lambda }+1\right )^{\frac {b \beta }{a \lambda }} {\mathrm e}^{-\frac {\textit {\_a} b \beta }{a}}}{\sqrt {2^{-\frac {b \beta }{a \lambda }} \left ({\mathrm e}^{2 \textit {\_a} \lambda }+1\right )^{\frac {b \beta }{a \lambda }} \RootOf \left (\beta y -\mathrm {arccosh}\left (\textit {\_Z} 2^{-\frac {b \beta }{a \lambda }} \left ({\mathrm e}^{2 \lambda x}+1\right )^{\frac {b \beta }{a \lambda }} {\mathrm e}^{-\frac {b \beta x}{a}}\right )\right ) {\mathrm e}^{-\frac {\textit {\_a} b \beta }{a}}-1}\, \sqrt {\left (\left ({\mathrm e}^{2 \textit {\_a} \lambda }+1\right )^{\frac {b \beta }{a \lambda }} \RootOf \left (\beta y -\mathrm {arccosh}\left (\textit {\_Z} 2^{-\frac {b \beta }{a \lambda }} \left ({\mathrm e}^{2 \lambda x}+1\right )^{\frac {b \beta }{a \lambda }} {\mathrm e}^{-\frac {b \beta x}{a}}\right )\right ) {\mathrm e}^{-\frac {\textit {\_a} b \beta }{a}}+2^{\frac {b \beta }{a \lambda }}\right ) 2^{-\frac {b \beta }{a \lambda }}}}d \textit {\_a} \right )}{a}}\]

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