7.6.10 4.4

7.6.10.1 [1484] Problem 1

problem number 1484

Added May 19, 2019.

Problem Chapter 6.4.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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7.6.10.2 [1485] Problem 2

problem number 1485

Added May 19, 2019.

Problem Chapter 6.4.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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7.6.10.3 [1486] Problem 3

problem number 1486

Added May 19, 2019.

Problem Chapter 6.4.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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7.6.10.4 [1487] Problem 4

problem number 1487

Added May 19, 2019.

Problem Chapter 6.4.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*coth(beta*y)*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))),output='realtime'));
 

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

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7.6.10.5 [1488] Problem 5

problem number 1488

Added May 19, 2019.

Problem Chapter 6.4.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

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

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7.6.10.6 [1489] Problem 6

problem number 1489

Added May 19, 2019.

Problem Chapter 6.4.4.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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