6.6.7 4.1

6.6.7.1 [1465] Problem 1

problem number 1465

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]];
 

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

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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},{\frac {za\lambda -\cosh \left ( \lambda \,x \right ) c}{a\lambda }} \right ) \]

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6.6.7.2 [1466] Problem 2

problem number 1466

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (z-\frac {c \cosh (\lambda x)}{a \lambda },\frac {\log \left (\tanh \left (\frac {\beta y}{2}\right )\right )}{\beta }-\frac {b x}{a}\right )\right \}\right \}\]

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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-bx\beta -2\,\arctanh \left ( {{\rm e}^{\beta \,y}} \right ) a}{b\beta }},{\frac {za\lambda -\cosh \left ( \lambda \,x \right ) c}{a\lambda }} \right ) \]

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6.6.7.3 [1467] Problem 3

problem number 1467

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\log \left (\tanh \left (\frac {\beta y}{2}\right )\right )}{\beta }-\frac {b x}{a},\frac {\log \left (\tanh \left (\frac {\gamma z}{2}\right )\right )}{\gamma }-\frac {c x}{a}\right )\right \}\right \}\]

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'));
 

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

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6.6.7.4 [1468] Problem 4

problem number 1468

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\log \left (\tanh \left (\frac {\beta y}{2}\right ) \tanh ^{-\frac {b \beta }{a \lambda }}\left (\frac {\lambda x}{2}\right )\right )}{\beta },\frac {\log \left (\tanh \left (\frac {\gamma z}{2}\right ) \tanh ^{-\frac {c \gamma }{a \lambda }}\left (\frac {\lambda x}{2}\right )\right )}{\gamma }\right )\right \}\right \}\]

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'));
 

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

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6.6.7.5 [1469] Problem 5

problem number 1469

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]];
 

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(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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-\cosh \left ( \lambda \,x \right ) b\beta +a\cosh \left ( \beta \,y \right ) \lambda }{b\beta \,\lambda }},{\frac {1}{c} \left ( -c\int ^{x}\!{\frac {1}{\sqrt {{\frac {\cosh \left ( {\it \_a}\,\lambda \right ) b\beta -\cosh \left ( \lambda \,x \right ) b\beta +\lambda \,a \left ( \cosh \left ( \beta \,y \right ) -1 \right ) }{a\lambda }}}}}{\frac {1}{\sqrt {{\frac {\cosh \left ( {\it \_a}\,\lambda \right ) b\beta -\cosh \left ( \lambda \,x \right ) b\beta +\lambda \,a \left ( \cosh \left ( \beta \,y \right ) +1 \right ) }{a\lambda }}}}}{d{\it \_a}}-16\,\arctanh \left ( {{\rm e}^{z/8}} \right ) a \right ) } \right ) \]

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6.6.7.6 [1470] Problem 6

problem number 1470

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]];
 

$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)*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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-\cosh \left ( \lambda \,x \right ) b\beta +a\cosh \left ( \beta \,y \right ) \lambda }{b\beta \,\lambda }},1/2\,{\frac {-32\,\arctanh \left ( {{\rm e}^{z/8}} \right ) a\lambda -c \left ( {{\rm e}^{-\lambda \,x}}+{{\rm e}^{\lambda \,x}} \right ) }{\lambda \,c}} \right ) \]

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