7.6.11 4.5

7.6.11.1 [1490] Problem 1

problem number 1490

Added May 19, 2019.

Problem Chapter 6.4.5.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*Sinh[lambda*x]*D[w[x, y,z], x] + b*Sinh[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]];
 

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

Maple ✓

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

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

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7.6.11.2 [1491] Problem 2

problem number 1491

Added May 19, 2019.

Problem Chapter 6.4.5.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*Sinh[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]];
 

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

Maple ✓

restart; 
pde :=  a*sinh(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'));
 

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

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7.6.11.3 [1492] Problem 3

problem number 1492

Added May 19, 2019.

Problem Chapter 6.4.5.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

$Aborted

Maple ✓

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

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

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7.6.11.4 [1493] Problem 4

problem number 1493

Added May 19, 2019.

Problem Chapter 6.4.5.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*Cosh[beta*y]*D[w[x, y,z], x] + b*Tanh[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]];
 

$Aborted

Maple ✓

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

\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {2 a \lambda \sinh \left (\beta y \right )+b \beta \ln \left (\tanh \left (\lambda x \right )-1\right )+b \beta \ln \left (\tanh \left (\lambda x \right )+1\right )}{2 b \beta \lambda }, \frac {a \arctan \left ({\mathrm e}^{\gamma z}\right )-\gamma c \left (\int _{}^{x}\frac {1}{\sqrt {\frac {b^{2} \beta ^{2} \ln \left (\tanh \left (\textit {\_a} \lambda \right )-1\right )^{2}+2 b^{2} \beta ^{2} \ln \left (\tanh \left (\textit {\_a} \lambda \right )-1\right ) \ln \left (\tanh \left (\textit {\_a} \lambda \right )+1\right )+b^{2} \beta ^{2} \ln \left (\tanh \left (\textit {\_a} \lambda \right )+1\right )^{2}+4 a^{2} \lambda ^{2}-2 \left (2 a \lambda \sinh \left (\beta y \right )+b \beta \ln \left (\tanh \left (\lambda x \right )-1\right )+b \beta \ln \left (\tanh \left (\lambda x \right )+1\right )\right ) b \beta \ln \left (\tanh \left (\textit {\_a} \lambda \right )-1\right )-2 \left (2 a \lambda \sinh \left (\beta y \right )+b \beta \ln \left (\tanh \left (\lambda x \right )-1\right )+b \beta \ln \left (\tanh \left (\lambda x \right )+1\right )\right ) b \beta \ln \left (\tanh \left (\textit {\_a} \lambda \right )+1\right )+\left (2 a \lambda \sinh \left (\beta y \right )+b \beta \ln \left (\tanh \left (\lambda x \right )-1\right )+b \beta \ln \left (\tanh \left (\lambda x \right )+1\right )\right )^{2}}{a^{2} \lambda ^{2}}}}d \textit {\_a} \right )}{\gamma c}\right )\]

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7.6.11.5 [1494] Problem 5

problem number 1494

Added May 19, 2019.

Problem Chapter 6.4.5.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*Coth[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*coth(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'));
 

\[w \left (x , y , z\right ) = c_{1} \left (\tanh \left (\gamma z \right )-1\right )^{-\frac {\textit {\_c}_{3}}{2 \gamma }} \left (\tanh \left (\gamma z \right )+1\right )^{-\frac {\textit {\_c}_{3}}{2 \gamma }} \left (\tanh ^{\frac {\textit {\_c}_{3}}{\gamma }}\left (\gamma z \right )\right ) \textit {\_F5} \left (\frac {a \ln \left (\RootOf \left (\beta y -\arcsinh \left (\textit {\_Z} 2^{-\frac {b \beta }{a \lambda }} \left ({\mathrm e}^{2 \lambda x}+1\right )^{\frac {b \beta }{a \lambda }} {\mathrm e}^{-\frac {b \beta x}{a}}\right )\right )\right )}{b \beta }\right ) {\mathrm e}^{-\frac {c \textit {\_c}_{3} 2^{-\frac {b \beta }{a \lambda }} \RootOf \left (\beta y -\arcsinh \left (\textit {\_Z} 2^{-\frac {b \beta }{a \lambda }} \left ({\mathrm e}^{2 \lambda x}+1\right )^{\frac {b \beta }{a \lambda }} {\mathrm e}^{-\frac {b \beta x}{a}}\right )\right ) \left (\int _{}^{x}\frac {\left ({\mathrm e}^{2 \textit {\_a} \lambda }+1\right )^{\frac {b \beta }{a \lambda }} {\mathrm e}^{-\frac {\textit {\_a} b \beta }{a}}}{\sqrt {4^{-\frac {b \beta }{a \lambda }} \left ({\mathrm e}^{2 \textit {\_a} \lambda }+1\right )^{\frac {2 b \beta }{a \lambda }} \RootOf \left (\beta y -\arcsinh \left (\textit {\_Z} 2^{-\frac {b \beta }{a \lambda }} \left ({\mathrm e}^{2 \lambda x}+1\right )^{\frac {b \beta }{a \lambda }} {\mathrm e}^{-\frac {b \beta x}{a}}\right )\right )^{2} {\mathrm e}^{-\frac {2 \textit {\_a} b \beta }{a}}+1}}d \textit {\_a} \right )}{a}}\]

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7.6.11.6 [1495] Problem 6

problem number 1495

Added May 19, 2019.

Problem Chapter 6.4.5.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*Coth[beta*y]*D[w[x, y,z], x] + b*Tanh[lambda*x]*D[w[x, y,z], y] +c*Coth[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*coth(beta*y)*diff(w(x,y,z),x)+ b*tanh(lambda*x)*diff(w(x,y,z),y)+c*coth(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'));
 

\[w \left (x , y , z\right ) = c_{1} \left (\coth \left (\gamma z \right )-1\right )^{-\frac {\textit {\_c}_{3}}{2 \gamma }} \left (\coth \left (\gamma z \right )+1\right )^{-\frac {\textit {\_c}_{3}}{2 \gamma }} \left (\coth ^{\frac {\textit {\_c}_{3}}{\gamma }}\left (\gamma z \right )\right ) \textit {\_F5} \left (\frac {a \ln \left (\RootOf \left (\beta y -\arcsinh \left (\textit {\_Z} 2^{-\frac {b \beta }{a \lambda }} \left ({\mathrm e}^{2 \lambda x}+1\right )^{\frac {b \beta }{a \lambda }} {\mathrm e}^{-\frac {b \beta x}{a}}\right )\right )\right )}{b \beta }\right ) {\mathrm e}^{-\frac {c \textit {\_c}_{3} 2^{-\frac {b \beta }{a \lambda }} \RootOf \left (\beta y -\arcsinh \left (\textit {\_Z} 2^{-\frac {b \beta }{a \lambda }} \left ({\mathrm e}^{2 \lambda x}+1\right )^{\frac {b \beta }{a \lambda }} {\mathrm e}^{-\frac {b \beta x}{a}}\right )\right ) \left (\int _{}^{x}\frac {\left ({\mathrm e}^{2 \textit {\_a} \lambda }+1\right )^{\frac {b \beta }{a \lambda }} {\mathrm e}^{-\frac {\textit {\_a} b \beta }{a}}}{\sqrt {4^{-\frac {b \beta }{a \lambda }} \left ({\mathrm e}^{2 \textit {\_a} \lambda }+1\right )^{\frac {2 b \beta }{a \lambda }} \RootOf \left (\beta y -\arcsinh \left (\textit {\_Z} 2^{-\frac {b \beta }{a \lambda }} \left ({\mathrm e}^{2 \lambda x}+1\right )^{\frac {b \beta }{a \lambda }} {\mathrm e}^{-\frac {b \beta x}{a}}\right )\right )^{2} {\mathrm e}^{-\frac {2 \textit {\_a} b \beta }{a}}+1}}d \textit {\_a} \right )}{a}}\]

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