6.6.9 4.3

6.6.9.1 [1477] Problem 1

problem number 1477

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'));
 

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

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6.6.9.2 [1478] Problem 2

problem number 1478

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'));
 

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

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6.6.9.3 [1479] Problem 3

problem number 1479

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'));
 

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

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6.6.9.4 [1480] Problem 4

problem number 1480

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'));
 

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

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6.6.9.5 [1481] Problem 5

problem number 1481

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'));
 

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

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6.6.9.6 [1482] Problem 6

problem number 1482

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'));
 

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

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