6.6.9 4.3

6.6.9.1 [1476] Problem 1

problem number 1476

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'));
 

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

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6.6.9.2 [1477] Problem 2

problem number 1477

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'));
 

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

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6.6.9.3 [1478] Problem 3

problem number 1478

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'));
 

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

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6.6.9.4 [1479] Problem 4

problem number 1479

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'));
 

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

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6.6.9.5 [1480] Problem 5

problem number 1480

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'));
 

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

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6.6.9.6 [1481] Problem 6

problem number 1481

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 ) ={{\it \_C1} \left ( \tanh \left ( \gamma \,z \right ) +1 \right ) ^{-{\frac {{\it \_c}_{{3}}}{2\,\gamma }}} \left ( \tanh \left ( \gamma \,z \right ) \right ) ^{{\frac {{\it \_c}_{{3}}}{\gamma }}} \left ( \tanh \left ( \gamma \,z \right ) -1 \right ) ^{-{\frac {{\it \_c}_{{3}}}{2\,\gamma }}}{\it \_F5} \left ( {\frac {a}{\beta \,b}\ln \left ( \RootOf \left ( \beta \,y-{\rm arccosh} \left ( \left ( {{\rm e}^{2\,x\lambda }}+1 \right ) ^{{\frac {\beta \,b}{a\lambda }}}{\it \_Z}\,{2}^{-{\frac {\beta \,b}{a\lambda }}}{{\rm e}^{-{\frac {b\beta \,x}{a}}}}\right ) \right ) \right ) } \right ) \left ( {{\rm e}^{{\frac {c{\it \_c}_{{3}}}{a}\RootOf \left ( \beta \,y-{\rm arccosh} \left ( \left ( {{\rm e}^{2\,x\lambda }}+1 \right ) ^{{\frac {\beta \,b}{a\lambda }}}{\it \_Z}\,{2}^{-{\frac {\beta \,b}{a\lambda }}}{{\rm e}^{-{\frac {b\beta \,x}{a}}}}\right ) \right ) \int ^{x}\!{ \left ( {{\rm e}^{2\,{\it \_a}\,\lambda }}+1 \right ) ^{{\frac {\beta \,b}{a\lambda }}}{{\rm e}^{-{\frac {{\it \_a}\,\beta \,b}{a}}}}{\frac {1}{\sqrt {{2}^{-{\frac {\beta \,b}{a\lambda }}} \left ( {{\rm e}^{2\,{\it \_a}\,\lambda }}+1 \right ) ^{{\frac {\beta \,b}{a\lambda }}}\RootOf \left ( \beta \,y-{\rm arccosh} \left ( \left ( {{\rm e}^{2\,x\lambda }}+1 \right ) ^{{\frac {\beta \,b}{a\lambda }}}{\it \_Z}\,{2}^{-{\frac {\beta \,b}{a\lambda }}}{{\rm e}^{-{\frac {b\beta \,x}{a}}}}\right ) \right ) {{\rm e}^{-{\frac {{\it \_a}\,\beta \,b}{a}}}}-1}}}{\frac {1}{\sqrt {{2}^{-{\frac {\beta \,b}{a\lambda }}} \left ( \left ( {{\rm e}^{2\,{\it \_a}\,\lambda }}+1 \right ) ^{{\frac {\beta \,b}{a\lambda }}}\RootOf \left ( \beta \,y-{\rm arccosh} \left ( \left ( {{\rm e}^{2\,x\lambda }}+1 \right ) ^{{\frac {\beta \,b}{a\lambda }}}{\it \_Z}\,{2}^{-{\frac {\beta \,b}{a\lambda }}}{{\rm e}^{-{\frac {b\beta \,x}{a}}}}\right ) \right ) {{\rm e}^{-{\frac {{\it \_a}\,\beta \,b}{a}}}}+{2}^{{\frac {\beta \,b}{a\lambda }}} \right ) }}}}{d{\it \_a}} \left ( {2}^{{\frac {\beta \,b}{a\lambda }}} \right ) ^{-1}}}} \right ) ^{-1}}\]

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