6.6.11 4.5

6.6.11.1 [1489] Problem 1

problem number 1489

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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-2\,\arctanh \left ( {{\rm e}^{\beta \,y}} \right ) a\lambda +2\,\arctanh \left ( {{\rm e}^{\lambda \,x}} \right ) b\beta }{b\beta \,\lambda }},{\frac {16\,\arctan \left ( {{\rm e}^{z/8}} \right ) a\lambda +2\,\arctanh \left ( {{\rm e}^{\lambda \,x}} \right ) c}{\lambda \,c}} \right ) \]

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6.6.11.2 [1490] Problem 2

problem number 1490

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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {2\,\arctan \left ( {{\rm e}^{\beta \,y}} \right ) a\lambda +2\,\arctanh \left ( {{\rm e}^{\lambda \,x}} \right ) b\beta }{b\beta \,\lambda }},{\frac {16\,\arctan \left ( {{\rm e}^{z/8}} \right ) a\lambda +2\,\arctanh \left ( {{\rm e}^{\lambda \,x}} \right ) c}{\lambda \,c}} \right ) \]

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6.6.11.3 [1491] Problem 3

problem number 1491

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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-\cosh \left ( \lambda \,x \right ) b\beta +a\cosh \left ( \beta \,y \right ) \lambda }{b\beta \,\lambda }},{\frac {16\,\arctan \left ( {{\rm e}^{z/8}} \right ) a\lambda -\cosh \left ( \lambda \,x \right ) c}{\lambda \,c}} \right ) \]

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6.6.11.4 [1492] Problem 4

problem number 1492

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]];
 

Failed

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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( 1/2\,{\frac {\ln \left ( \tanh \left ( \lambda \,x \right ) -1 \right ) b\beta +\ln \left ( \tanh \left ( \lambda \,x \right ) +1 \right ) b\beta +2\,a\sinh \left ( \beta \,y \right ) \lambda }{b\beta \,\lambda }},{\frac {1}{c} \left ( 8\,\arctan \left ( {{\rm e}^{z/8}} \right ) a-c\int ^{x}\!{\frac {1}{\sqrt {{\frac { \left ( \ln \left ( \tanh \left ( \lambda \,x \right ) -1 \right ) b\beta +\ln \left ( \tanh \left ( \lambda \,x \right ) +1 \right ) b\beta +2\,a\sinh \left ( \beta \,y \right ) \lambda \right ) ^{2}-2\,\ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) -1 \right ) \left ( \ln \left ( \tanh \left ( \lambda \,x \right ) -1 \right ) b\beta +\ln \left ( \tanh \left ( \lambda \,x \right ) +1 \right ) b\beta +2\,a\sinh \left ( \beta \,y \right ) \lambda \right ) b\beta -2\,\ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) +1 \right ) \left ( \ln \left ( \tanh \left ( \lambda \,x \right ) -1 \right ) b\beta +\ln \left ( \tanh \left ( \lambda \,x \right ) +1 \right ) b\beta +2\,a\sinh \left ( \beta \,y \right ) \lambda \right ) b\beta + \left ( \ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) -1 \right ) \right ) ^{2}{b}^{2}{\beta }^{2}+2\,\ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) -1 \right ) \ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) +1 \right ) {b}^{2}{\beta }^{2}+ \left ( \ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) +1 \right ) \right ) ^{2}{b}^{2}{\beta }^{2}+4\,{\lambda }^{2}{a}^{2}}{{\lambda }^{2}{a}^{2}}}}}}{d{\it \_a}} \right ) } \right ) \]

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6.6.11.5 [1493] Problem 5

problem number 1493

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'));
 

\[w \left ( x,y,z \right ) ={\frac {{\it \_C1}\, \left ( \left ( \tanh \left ( z/8 \right ) \right ) ^{{\it \_c}_{{3}}} \right ) ^{8}}{ \left ( \left ( \tanh \left ( z/8 \right ) +1 \right ) ^{{\it \_c}_{{3}}} \right ) ^{4} \left ( \left ( \tanh \left ( z/8 \right ) -1 \right ) ^{{\it \_c}_{{3}}} \right ) ^{4}}{\it \_F5} \left ( {\frac {a}{b\beta }\ln \left ( \RootOf \left ( \beta \,y-\arcsinh \left ( \left ( {{\rm e}^{2\,\lambda \,x}}+1 \right ) ^{{\frac {b\beta }{a\lambda }}}{\it \_Z}\,{2}^{-{\frac {b\beta }{a\lambda }}}{{\rm e}^{-{\frac {bx\beta }{a}}}} \right ) \right ) \right ) } \right ) \left ( {{\rm e}^{{\frac {c{\it \_c}_{{3}}}{a}\RootOf \left ( \beta \,y-\arcsinh \left ( \left ( {{\rm e}^{2\,\lambda \,x}}+1 \right ) ^{{\frac {b\beta }{a\lambda }}}{\it \_Z}\,{2}^{-{\frac {b\beta }{a\lambda }}}{{\rm e}^{-{\frac {bx\beta }{a}}}} \right ) \right ) \int ^{x}\! \left ( {{\rm e}^{2\,{\it \_a}\,\lambda }}+1 \right ) ^{{\frac {b\beta }{a\lambda }}}{{\rm e}^{-{\frac {{\it \_a}\,b\beta }{a}}}}{\frac {1}{\sqrt {1+{4}^{-{\frac {b\beta }{a\lambda }}} \left ( {{\rm e}^{2\,{\it \_a}\,\lambda }}+1 \right ) ^{2\,{\frac {b\beta }{a\lambda }}} \left ( \RootOf \left ( \beta \,y-\arcsinh \left ( \left ( {{\rm e}^{2\,\lambda \,x}}+1 \right ) ^{{\frac {b\beta }{a\lambda }}}{\it \_Z}\,{2}^{-{\frac {b\beta }{a\lambda }}}{{\rm e}^{-{\frac {bx\beta }{a}}}} \right ) \right ) \right ) ^{2}{{\rm e}^{-2\,{\frac {{\it \_a}\,b\beta }{a}}}}}}}{d{\it \_a}} \left ( {2}^{{\frac {b\beta }{a\lambda }}} \right ) ^{-1}}}} \right ) ^{-1}}\]

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6.6.11.6 [1494] Problem 6

problem number 1494

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'));
 

\[w \left ( x,y,z \right ) ={\frac {{\it \_C1}\, \left ( \left ( {\rm coth} \left (z/8\right ) \right ) ^{{\it \_c}_{{3}}} \right ) ^{8}}{ \left ( \left ( {\rm coth} \left (z/8\right )-1 \right ) ^{{\it \_c}_{{3}}} \right ) ^{4} \left ( \left ( {\rm coth} \left (z/8\right )+1 \right ) ^{{\it \_c}_{{3}}} \right ) ^{4}}{\it \_F5} \left ( {\frac {a}{b\beta }\ln \left ( \RootOf \left ( \beta \,y-\arcsinh \left ( \left ( {{\rm e}^{2\,\lambda \,x}}+1 \right ) ^{{\frac {b\beta }{a\lambda }}}{\it \_Z}\,{2}^{-{\frac {b\beta }{a\lambda }}}{{\rm e}^{-{\frac {bx\beta }{a}}}} \right ) \right ) \right ) } \right ) \left ( {{\rm e}^{{\frac {c{\it \_c}_{{3}}}{a}\RootOf \left ( \beta \,y-\arcsinh \left ( \left ( {{\rm e}^{2\,\lambda \,x}}+1 \right ) ^{{\frac {b\beta }{a\lambda }}}{\it \_Z}\,{2}^{-{\frac {b\beta }{a\lambda }}}{{\rm e}^{-{\frac {bx\beta }{a}}}} \right ) \right ) \int ^{x}\! \left ( {{\rm e}^{2\,{\it \_a}\,\lambda }}+1 \right ) ^{{\frac {b\beta }{a\lambda }}}{{\rm e}^{-{\frac {{\it \_a}\,b\beta }{a}}}}{\frac {1}{\sqrt {1+{4}^{-{\frac {b\beta }{a\lambda }}} \left ( {{\rm e}^{2\,{\it \_a}\,\lambda }}+1 \right ) ^{2\,{\frac {b\beta }{a\lambda }}} \left ( \RootOf \left ( \beta \,y-\arcsinh \left ( \left ( {{\rm e}^{2\,\lambda \,x}}+1 \right ) ^{{\frac {b\beta }{a\lambda }}}{\it \_Z}\,{2}^{-{\frac {b\beta }{a\lambda }}}{{\rm e}^{-{\frac {bx\beta }{a}}}} \right ) \right ) \right ) ^{2}{{\rm e}^{-2\,{\frac {{\it \_a}\,b\beta }{a}}}}}}}{d{\it \_a}} \left ( {2}^{{\frac {b\beta }{a\lambda }}} \right ) ^{-1}}}} \right ) ^{-1}}\]

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