6.6.7 4.1

6.6.7.1 [1464] Problem 1

problem number 1464

Added May 19, 2019.

Problem Chapter 6.4.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a w_x + b w_y + c \sinh (\lambda x) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*D[w[x, y,z], y] +c*Sinh[lambda*x]*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},z-\int _1^x\frac {c \sinh (\lambda K[1])}{a}dK[1]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+c*sinh(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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {y a -b x}{a}, \frac {z a \lambda -c \cosh \left (\lambda x \right )}{a \lambda }\right )\]

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6.6.7.2 [1465] Problem 2

problem number 1465

Added May 19, 2019.

Problem Chapter 6.4.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a w_x + b \sinh (\beta y) w_y + c \sinh (\lambda x) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*Sinh[beta*y]*D[w[x, y,z], y] +c*Sinh[lambda*x]*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {b x}{a}-\frac {\text {arctanh}(\cosh (\beta y))}{\beta },z-\int _1^x\frac {c \sinh (\lambda K[1])}{a}dK[1]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*sinh(beta*y)*diff(w(x,y,z),y)+c*sinh(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'));
 

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

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6.6.7.3 [1466] Problem 3

problem number 1466

Added May 19, 2019.

Problem Chapter 6.4.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a w_x + b \sinh (\beta y) w_y + c \sinh (\gamma z) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*Sinh[beta*y]*D[w[x, y,z], y] +c*Sinh[gamma*z]*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*sinh(beta*y)*diff(w(x,y,z),y)+c*sinh(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'));
 

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

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6.6.7.4 [1467] Problem 4

problem number 1467

Added May 19, 2019.

Problem Chapter 6.4.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a \sinh (\lambda x) w_x + b \sinh (\beta y) w_y + c \sinh (\gamma z) w_z = 0 \]

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*Sinh[gamma*z]*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  a*sinh(lambda*x)*diff(w(x,y,z),x)+ b*sinh(beta*y)*diff(w(x,y,z),y)+c*sinh(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'));
 

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

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6.6.7.5 [1468] Problem 5

problem number 1468

Added May 19, 2019.

Problem Chapter 6.4.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a \sinh (\beta y) w_x + b \sinh (\lambda x) w_y + c \sinh (\gamma z) w_z = 0 \]

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[gamma*z]*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {\gamma \int _1^x\frac {c \text {csch}\left (\beta \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\sinh (\beta K[1])dK[1]\& \right ]\left [\int _1^y\sinh (\beta K[1])dK[1]-\int _1^x\frac {b \sinh (\lambda K[2])}{a}dK[2]+\int _1^{K[3]}\frac {b \sinh (\lambda K[2])}{a}dK[2]\right ]\right )}{a}dK[3]+\text {arctanh}(\cosh (\gamma z))}{\gamma },\int _1^y\sinh (\beta K[1])dK[1]-\int _1^x\frac {b \sinh (\lambda K[2])}{a}dK[2]\right )\right \}\right \}\]

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(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'));
 

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

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6.6.7.6 [1469] Problem 6

problem number 1469

Added May 19, 2019.

Problem Chapter 6.4.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a \sinh (\beta y) w_x + b \sinh (\lambda x) w_y + c \sinh (\lambda x) \sinh (\beta y) \sinh (\gamma z) w_z = 0 \]

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]*Sinh[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)*sinh(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'));
 

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

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