6.6.11 4.5

6.6.11.1 [1488] Problem 1

problem number 1488

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {c \text {arctanh}(\cosh (\lambda x))}{a \lambda }-\frac {\cot ^{-1}(\sinh (\gamma z))}{\gamma },\frac {b \text {arctanh}(\cosh (\lambda x))}{a \lambda }-\frac {\text {arctanh}(\cosh (\beta y))}{\beta }\right )\right \}\right \}\]

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-\ln \left (\tanh \left (\frac {\lambda x}{2}\right )\right ) b \beta -2 a \,\operatorname {arctanh}\left ({\mathrm e}^{\beta y}\right ) \lambda }{\lambda b \beta }, \frac {2 a \arctan \left ({\mathrm e}^{\gamma z}\right ) \lambda -\ln \left (\tanh \left (\frac {\lambda x}{2}\right )\right ) c \gamma }{\lambda c \gamma }\right )\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.6.11.2 [1489] Problem 2

problem number 1489

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b \text {arctanh}(\cosh (\lambda x))}{a \lambda }-\frac {\cot ^{-1}(\sinh (\beta y))}{\beta },\frac {c \text {arctanh}(\cosh (\lambda x))}{a \lambda }-\frac {\cot ^{-1}(\sinh (\gamma z))}{\gamma }\right )\right \}\right \}\]

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {2 a \arctan \left ({\mathrm e}^{\beta y}\right ) \lambda -\ln \left (\tanh \left (\frac {\lambda x}{2}\right )\right ) b \beta }{\lambda b \beta }, \frac {2 a \arctan \left ({\mathrm e}^{\gamma z}\right ) \lambda -\ln \left (\tanh \left (\frac {\lambda x}{2}\right )\right ) c \gamma }{\lambda c \gamma }\right )\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.6.11.3 [1490] Problem 3

problem number 1490

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]];
 

Failed

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {a \cosh \left (\beta y \right ) \lambda -\cosh \left (\lambda x \right ) b \beta }{\lambda b \beta }, \frac {2 a \arctan \left ({\mathrm e}^{\gamma z}\right ) \lambda -\cosh \left (\lambda x \right ) c \gamma }{\lambda c \gamma }\right )\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.6.11.4 [1491] Problem 4

problem number 1491

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]];
 

\begin{align*}& \left \{w(x,y,z)\to c_1\left (\int _1^y\cosh (\beta K[1])dK[1]-\frac {b \log (\cosh (\lambda x))}{a \lambda },-\frac {\gamma \int _1^x\frac {c \text {sech}\left (\beta \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\cosh (\beta K[1])dK[1]\&\right ]\left [\frac {b (\log (\cosh (\lambda K[2]))-\log (\cosh (\lambda x)))}{a \lambda }+\int _1^y\cosh (\beta K[1])dK[1]\right ]\right )}{a}dK[2]+\cot ^{-1}(\sinh (\gamma z))}{\gamma }\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (\int _1^y\cosh (\beta K[1])dK[1]-\frac {b \log (\cosh (\lambda x))}{a \lambda },\frac {\cot ^{-1}(\sinh (\gamma z))}{\gamma }-\int _1^x\frac {c \text {sech}\left (\beta \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\cosh (\beta K[1])dK[1]\&\right ]\left [\frac {b (\log (\cosh (\lambda K[2]))-\log (\cosh (\lambda x)))}{a \lambda }+\int _1^y\cosh (\beta K[1])dK[1]\right ]\right )}{a}dK[2]\right )\right \}\\\end{align*}

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-\ln \left (\cosh \left (\lambda x \right )\right ) b \beta +a \sinh \left (\beta y \right ) \lambda }{\lambda b \beta }, \frac {-c \gamma \int _{}^{x}\frac {1}{\sqrt {\frac {\ln \left (\cosh \left (\lambda \textit {\_a} \right )\right )^{2} b^{2} \beta ^{2}-2 \ln \left (\cosh \left (\lambda \textit {\_a} \right )\right ) \ln \left (\cosh \left (\lambda x \right )\right ) b^{2} \beta ^{2}+2 \ln \left (\cosh \left (\lambda \textit {\_a} \right )\right ) \sinh \left (\beta y \right ) a b \beta \lambda +\ln \left (\cosh \left (\lambda x \right )\right )^{2} b^{2} \beta ^{2}-2 \ln \left (\cosh \left (\lambda x \right )\right ) \sinh \left (\beta y \right ) a b \beta \lambda +a^{2} \cosh \left (\beta y \right )^{2} \lambda ^{2}}{a^{2} \lambda ^{2}}}}d \textit {\_a} +2 a \arctan \left ({\mathrm e}^{\gamma z}\right )}{c \gamma }\right )\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.6.11.5 [1492] Problem 5

problem number 1492

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-b \beta \ln \left ({\mathrm e}^{2 \lambda x}+1\right )+\ln \left (i \sinh \left (\beta y \right )\right ) a \lambda +b \beta \left (\lambda x +\ln \left (2\right )\right )}{\lambda b \beta }, \frac {\ln \left (\frac {\tanh \left (\gamma z \right )}{\sqrt {-{\mathrm e}^{\frac {2 \int _{}^{x}\frac {1}{\sqrt {1+\operatorname {csch}\left (\beta y \right )^{2} {\mathrm e}^{\frac {2 b \beta \left (-x +\textit {\_a} \right )}{a}} \left ({\mathrm e}^{2 \lambda x}+1\right )^{\frac {2 b \beta }{a \lambda }} \left ({\mathrm e}^{2 \lambda \textit {\_a}}+1\right )^{-\frac {2 b \beta }{a \lambda }}}}d \textit {\_a} c \gamma }{a}} \operatorname {sech}\left (\gamma z \right )^{2}}}\right ) a}{c \gamma }\right )\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.6.11.6 [1493] Problem 6

problem number 1493

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-b \beta \ln \left ({\mathrm e}^{2 \lambda x}+1\right )+\ln \left (i \sinh \left (\beta y \right )\right ) \lambda a +b \beta \left (\lambda x +\ln \left (2\right )\right )}{\lambda b \beta }, \frac {-2 \int _{}^{x}\frac {1}{\sqrt {1+\operatorname {csch}\left (\beta y \right )^{2} {\mathrm e}^{\frac {2 b \beta \left (-x +\textit {\_a} \right )}{a}} \left ({\mathrm e}^{2 \lambda x}+1\right )^{\frac {2 b \beta }{\lambda a}} \left ({\mathrm e}^{2 \lambda \textit {\_a}}+1\right )^{-\frac {2 b \beta }{\lambda a}}}}d \textit {\_a} c \gamma -a \left (\ln \left (\coth \left (\gamma z \right )+1\right )-\ln \left (\frac {\coth \left (\gamma z \right )^{2}}{\coth \left (\gamma z \right )-1}\right )\right )}{2 c \gamma }\right )\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________