6.6.10 4.4

6.6.10.1 [1483] Problem 1

problem number 1483

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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},{\frac {1}{c} \left ( 4\,a\ln \left ( {\frac { \left ( \RootOf \left ( -8\,{\rm arccoth} \left ({\it \_Z}-1\right )+z \right ) -1 \right ) ^{2}}{\RootOf \left ( -8\,{\rm arccoth} \left ({\it \_Z}-1\right )+z \right ) -2}} \right ) -4\,\ln \left ( \RootOf \left ( -8\,{\rm arccoth} \left ({\it \_Z}-1\right )+z \right ) \right ) a-cx \right ) } \right ) \]

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6.6.10.2 [1484] Problem 2

problem number 1484

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'));
 

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

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6.6.10.3 [1485] Problem 3

problem number 1485

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'));
 

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

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6.6.10.4 [1486] Problem 4

problem number 1486

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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -1/2\,{\frac {1}{b\beta } \left ( 2\,bx\beta +\ln \left ( \RootOf \left ( \beta \,y-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) \right ) a-a\ln \left ( {\frac { \left ( \RootOf \left ( \beta \,y-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) -1 \right ) ^{2}}{\RootOf \left ( \beta \,y-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) -2}} \right ) \right ) },{\frac {1}{c} \left ( 4\,a\ln \left ( {\frac { \left ( \RootOf \left ( -8\,{\rm arccoth} \left ({\it \_Z}-1\right )+z \right ) -1 \right ) ^{2}}{\RootOf \left ( -8\,{\rm arccoth} \left ({\it \_Z}-1\right )+z \right ) -2}} \right ) -4\,\ln \left ( \RootOf \left ( -8\,{\rm arccoth} \left ({\it \_Z}-1\right )+z \right ) \right ) a-cx \right ) } \right ) \]

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6.6.10.5 [1487] Problem 5

problem number 1487

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'));
 

\[w \left ( x,y,z \right ) ={{\it \_C3}\,{\it \_C2}\,{\it \_C1} \left ( {\rm coth} \left (\lambda \,x\right )-1 \right ) ^{-1/2\,{\frac {{\it \_c}_{{1}}}{\lambda }}} \left ( {\rm coth} \left (\lambda \,x\right )+1 \right ) ^{-1/2\,{\frac {{\it \_c}_{{1}}}{\lambda }}} \left ( {\rm coth} \left (\lambda \,x\right ) \right ) ^{{\frac {{\it \_c}_{{1}}}{\lambda }}} \left ( {\rm coth} \left (\beta \,y\right )-1 \right ) ^{-1/2\,{\frac {{\it \_c}_{{2}}}{\beta }}} \left ( {\rm coth} \left (\beta \,y\right )+1 \right ) ^{-1/2\,{\frac {{\it \_c}_{{2}}}{\beta }}} \left ( {\rm coth} \left (\beta \,y\right ) \right ) ^{{\frac {{\it \_c}_{{2}}}{\beta }}} \left ( \left ( {\rm coth} \left (z/8\right )+1 \right ) ^{{\frac {a{\it \_c}_{{1}}}{c}}} \right ) ^{4} \left ( \left ( {\rm coth} \left (z/8\right )+1 \right ) ^{{\frac {b{\it \_c}_{{2}}}{c}}} \right ) ^{4} \left ( \left ( {\rm coth} \left (z/8\right )-1 \right ) ^{{\frac {a{\it \_c}_{{1}}}{c}}} \right ) ^{4} \left ( \left ( {\rm coth} \left (z/8\right )-1 \right ) ^{{\frac {b{\it \_c}_{{2}}}{c}}} \right ) ^{4} \left ( \left ( {\rm coth} \left (z/8\right ) \right ) ^{{\frac {a{\it \_c}_{{1}}}{c}}} \right ) ^{-8} \left ( \left ( {\rm coth} \left (z/8\right ) \right ) ^{{\frac {b{\it \_c}_{{2}}}{c}}} \right ) ^{-8}}\]

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6.6.10.6 [1488] Problem 6

problem number 1488

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'));
 

\[{\it PDESolStruc} \left ( w \left ( x,y,z \right ) =-4\,{\frac {{\it \_c}_{{3}}\ln \left ( {\rm coth} \left (z/8\right )-1 \right ) }{c}}-4\,{\frac {{\it \_c}_{{3}}\ln \left ( {\rm coth} \left (z/8\right )+1 \right ) }{c}}+8\,{\frac {{\it \_c}_{{3}}\ln \left ( {\rm coth} \left (z/8\right ) \right ) }{c}}+{\it \_C1}+{\it \_F4} \left ( x,y \right ) ,[ \left \{ a{\rm coth} \left (\beta \,y\right ){\frac {\partial }{\partial x}}{\it \_F4} \left ( x,y \right ) +b{\rm coth} \left (\lambda \,x\right ){\frac {\partial }{\partial y}}{\it \_F4} \left ( x,y \right ) +{\it \_c}_{{3}}=0 \right \} ] \right ) \]

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