6.6.8 4.2

6.6.8.1 [1471] Problem 1

problem number 1471

Added May 19, 2019.

Problem Chapter 6.4.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.6.8.2 [1472] Problem 2

problem number 1472

Added May 19, 2019.

Problem Chapter 6.4.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.6.8.3 [1473] Problem 3

problem number 1473

Added May 19, 2019.

Problem Chapter 6.4.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {2 \tan ^{-1}\left (\tanh \left (\frac {\beta y}{2}\right )\right )}{\beta }-\frac {b x}{a},z-\frac {c \sinh (\lambda x)}{a \lambda }\right )\right \}\right \}\]

Maple ✓

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

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

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6.6.8.4 [1474] Problem 4

problem number 1474

Added May 19, 2019.

Problem Chapter 6.4.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {2 \tan ^{-1}\left (\tanh \left (\frac {\beta y}{2}\right )\right )}{\beta }-\frac {b x}{a},\frac {2 \tan ^{-1}\left (\tanh \left (\frac {\gamma z}{2}\right )\right )}{\gamma }-\frac {c x}{a}\right )\right \}\right \}\]

Maple ✓

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

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

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6.6.8.5 [1475] Problem 5

problem number 1475

Added May 19, 2019.

Problem Chapter 6.4.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*Cosh[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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {2 \tan ^{-1}\left (\tanh \left (\frac {\beta y}{2}\right )\right )}{\beta }-\frac {2 b \tan ^{-1}\left (\tanh \left (\frac {\lambda x}{2}\right )\right )}{a \lambda },\frac {2 \tan ^{-1}\left (\tanh \left (\frac {\gamma z}{2}\right )\right )}{\gamma }-\frac {2 c \tan ^{-1}\left (\tanh \left (\frac {\lambda x}{2}\right )\right )}{a \lambda }\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*cosh(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'));
 

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

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6.6.8.6 [1476] Problem 6

problem number 1476

Added May 19, 2019.

Problem Chapter 6.4.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*Cosh[beta*y]*D[w[x, y,z], x] + b*Cosh[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*cosh(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 ( {\frac {a\sinh \left ( \beta \,y \right ) \lambda -\sinh \left ( \lambda \,x \right ) b\beta }{b\beta \,\lambda }},16\,{\frac {1}{\sqrt { \left ( -\sinh \left ( \lambda \,x \right ) +i \right ) \left ( \sinh \left ( \lambda \,x \right ) +i \right ) \left ( i-\sinh \left ( \beta \,y \right ) \right ) \left ( i+\sinh \left ( \beta \,y \right ) \right ) }\sqrt { \left ( \sinh \left ( \beta \,y \right ) \right ) ^{2}+1}c\lambda \,\cosh \left ( \lambda \,x \right ) \left ( a\sinh \left ( \beta \,y \right ) \lambda -\sinh \left ( \lambda \,x \right ) b\beta +ia\lambda -ib\beta \right ) } \left ( -1/4\,\sqrt {{\frac {ia\lambda \, \left ( i-\sinh \left ( \beta \,y \right ) \right ) }{ \left ( -a\sinh \left ( \beta \,y \right ) \lambda +\sinh \left ( \lambda \,x \right ) b\beta +ia\lambda -ib\beta \right ) \left ( \sinh \left ( \lambda \,x \right ) +i \right ) }}} \left ( \left ( \left ( \left ( 1/2\,a\lambda -1/2\,b\beta \right ) \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}+\lambda \,a \left ( i+\sinh \left ( \beta \,y \right ) \right ) \sinh \left ( \lambda \,x \right ) -1/2\,a\lambda -1/2\,b\beta \right ) \sqrt {{\frac {i \left ( i+\sinh \left ( \beta \,y \right ) \right ) \lambda \,a}{ \left ( a\sinh \left ( \beta \,y \right ) \lambda -\sinh \left ( \lambda \,x \right ) b\beta +ia\lambda +ib\beta \right ) \left ( \sinh \left ( \lambda \,x \right ) +i \right ) }}}+1/2\,\sqrt {{\frac {-i \left ( i+\sinh \left ( \beta \,y \right ) \right ) \lambda \,a}{ \left ( a\sinh \left ( \beta \,y \right ) \lambda -\sinh \left ( \lambda \,x \right ) b\beta +ia\lambda +ib\beta \right ) \left ( \sinh \left ( \lambda \,x \right ) +i \right ) }}}\sinh \left ( \lambda \,x \right ) b\beta \right ) \sqrt {-{\frac { \left ( a\sinh \left ( \beta \,y \right ) \lambda -\sinh \left ( \lambda \,x \right ) b\beta +ia\lambda -ib\beta \right ) \left ( -\sinh \left ( \lambda \,x \right ) +i \right ) }{ \left ( a\sinh \left ( \beta \,y \right ) \lambda -\sinh \left ( \lambda \,x \right ) b\beta +ia\lambda +ib\beta \right ) \left ( \sinh \left ( \lambda \,x \right ) +i \right ) }}}-1/2\,\sqrt {{\frac { \left ( a\sinh \left ( \beta \,y \right ) \lambda -\sinh \left ( \lambda \,x \right ) b\beta +ia\lambda -ib\beta \right ) \left ( -\sinh \left ( \lambda \,x \right ) +i \right ) }{ \left ( a\sinh \left ( \beta \,y \right ) \lambda -\sinh \left ( \lambda \,x \right ) b\beta +ia\lambda +ib\beta \right ) \left ( \sinh \left ( \lambda \,x \right ) +i \right ) }}}\sqrt {{\frac {i \left ( i+\sinh \left ( \beta \,y \right ) \right ) \lambda \,a}{ \left ( a\sinh \left ( \beta \,y \right ) \lambda -\sinh \left ( \lambda \,x \right ) b\beta +ia\lambda +ib\beta \right ) \left ( \sinh \left ( \lambda \,x \right ) +i \right ) }}} \left ( \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}\sinh \left ( \beta \,y \right ) a\lambda - \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{3}b\beta -a\sinh \left ( \beta \,y \right ) \lambda \right ) \right ) \sqrt { \left ( \left ( \sinh \left ( \beta \,y \right ) \right ) ^{2}+1 \right ) \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}c\EllipticF \left ( \sqrt {-{\frac { \left ( a\sinh \left ( \beta \,y \right ) \lambda -\sinh \left ( \lambda \,x \right ) b\beta +ia\lambda -ib\beta \right ) \left ( -\sinh \left ( \lambda \,x \right ) +i \right ) }{ \left ( a\sinh \left ( \beta \,y \right ) \lambda -\sinh \left ( \lambda \,x \right ) b\beta +ia\lambda +ib\beta \right ) \left ( \sinh \left ( \lambda \,x \right ) +i \right ) }}},\sqrt {-{\frac { \left ( a\sinh \left ( \beta \,y \right ) \lambda -\sinh \left ( \lambda \,x \right ) b\beta \right ) ^{2}+{\lambda }^{2}{a}^{2}+2\,b\beta \,a\lambda +{b}^{2}{\beta }^{2}}{ \left ( -a\sinh \left ( \beta \,y \right ) \lambda +\sinh \left ( \lambda \,x \right ) b\beta +ia\lambda -ib\beta \right ) \left ( a\sinh \left ( \beta \,y \right ) \lambda -\sinh \left ( \lambda \,x \right ) b\beta +ia\lambda -ib\beta \right ) }}} \right ) +\lambda \,\sqrt { \left ( \sinh \left ( \beta \,y \right ) \right ) ^{2}+1}\sqrt { \left ( -\sinh \left ( \lambda \,x \right ) +i \right ) \left ( \sinh \left ( \lambda \,x \right ) +i \right ) \left ( i-\sinh \left ( \beta \,y \right ) \right ) \left ( i+\sinh \left ( \beta \,y \right ) \right ) } \left ( a\sinh \left ( \beta \,y \right ) \lambda -\sinh \left ( \lambda \,x \right ) b\beta +ia\lambda -ib\beta \right ) a\cosh \left ( \lambda \,x \right ) \arctan \left ( {{\rm e}^{z/8}} \right ) \right ) } \right ) \]

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