6.4.12 4.5

6.4.12.1 [1096] Problem 1

problem number 1096

Added Feb. 23, 2019.

Problem Chapter 4.4.5.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = (c \sinh (\lambda x) + k \cosh (\mu y)) w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) e^{\frac {c \cosh (\lambda x)}{a \lambda }+\frac {k \sinh (\mu y)}{b \mu }}\right \}\right \}\]

Maple ✓

restart; 
pde := a*diff(w(x,y),x)+b*diff(w(x,y),y) =   (c*sinh(lambda*x) + k*cosh(mu*y))*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) {{\rm e}^{{\frac {ak\lambda \,\sinh \left ( \mu \,y \right ) +\cosh \left ( x\lambda \right ) c\mu \,b}{a\lambda \,\mu \,b}}}}\]

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6.4.12.2 [1097] Problem 2

problem number 1097

Added Feb. 23, 2019.

Problem Chapter 4.4.5.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = (\tanh (\lambda x)+k \coth (\mu y)) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == (Tanh[lambda*x] + k*Coth[mu*y])*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \sqrt [a \lambda ]{\cosh (\lambda x)} c_1\left (y-\frac {b x}{a}\right ) e^{\frac {k (\log (\tanh (\mu y))+\log (\cosh (\mu y)))}{b \mu }}\right \}\right \}\]

Maple ✓

restart; 
pde := a*diff(w(x,y),x)+b*diff(w(x,y),y) =   (tanh(lambda*x)+k*coth(mu*y))*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \left ( \cosh \left ( x\lambda \right ) \right ) ^{{\frac {1}{a\lambda }}} \left ( \sinh \left ( \mu \,y \right ) \right ) ^{{\frac {k}{\mu \,b}}}\]

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6.4.12.3 [1098] Problem 3

problem number 1098

Added Feb. 23, 2019.

Problem Chapter 4.4.5.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + a \sinh (\mu y) w_y = b \cosh (\lambda x) w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to e^{\frac {b \sinh (\lambda x)}{\lambda }} c_1\left (\frac {\log \left (\tanh \left (\frac {\mu y}{2}\right )\right )}{\mu }-a x\right )\right \}\right \}\]

Maple ✓

restart; 
pde := diff(w(x,y),x)+a*sinh(mu*y)*diff(w(x,y),y) =   b*cosh(lambda*x)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-xa\mu -2\,\arctanh \left ( {{\rm e}^{\mu \,y}} \right ) }{a\mu }} \right ) {{\rm e}^{{\frac {b\sinh \left ( x\lambda \right ) }{\lambda }}}}\]

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6.4.12.4 [1099] Problem 4

problem number 1099

Added Feb. 23, 2019.

Problem Chapter 4.4.5.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + a \sinh (\mu y) w_y = b \tanh (\lambda x) w \]

Mathematica ✓

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

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

Maple ✓

restart; 
pde := diff(w(x,y),x)+a*sinh(mu*y)*diff(w(x,y),y) =   b*tanh(lambda*x)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-xa\mu -2\,\arctanh \left ( {{\rm e}^{\mu \,y}} \right ) }{a\mu }} \right ) \left ( \tanh \left ( x\lambda \right ) -1 \right ) ^{-{\frac {b}{2\,\lambda }}} \left ( \tanh \left ( x\lambda \right ) +1 \right ) ^{-{\frac {b}{2\,\lambda }}}\]

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6.4.12.5 [1100] Problem 5

problem number 1100

Added Feb. 23, 2019.

Problem Chapter 4.4.5.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a \sinh (\lambda x) w_x + b \cosh (\mu y) w_y = w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \sqrt [a \lambda ]{\tanh \left (\frac {\lambda x}{2}\right )} c_1\left (\frac {2 \tan ^{-1}\left (\tanh \left (\frac {\mu y}{2}\right )\right )}{\mu }-\frac {b \log \left (\tanh \left (\frac {\lambda x}{2}\right )\right )}{a \lambda }\right )\right \}\right \}\]

Maple ✓

restart; 
pde := a*sinh(lambda*x)*diff(w(x,y),x)+b*cosh(mu*y)^n*diff(w(x,y),y) =  w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{\lambda } \left ( \int \!{\frac { \left ( \cosh \left ( \mu \,y \right ) \right ) ^{-n}a}{b}}\,{\rm d}y\lambda +2\,\arctanh \left ( {{\rm e}^{x\lambda }} \right ) \right ) } \right ) {{\rm e}^{-2\,{\frac {\arctanh \left ( {{\rm e}^{x\lambda }} \right ) }{a\lambda }}}}\]

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6.4.12.6 [1101] Problem 6

problem number 1101

Added Feb. 23, 2019.

Problem Chapter 4.4.5.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a \tanh (\lambda x) w_x + b \coth (\mu y) w_y = w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*Tanh[lambda*x]*D[w[x, y], x] + b*Coth[mu*y]*D[w[x, y], y] == w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \sqrt [a \lambda ]{\sinh (\lambda x)} c_1\left (-\frac {2 a \cosh (\mu y) \sinh ^{-\frac {b \mu }{a \lambda }}(\lambda x)}{\mu }\right )\right \}\right \}\]

Maple ✓

restart; 
pde := a*tanh(lambda*x)*diff(w(x,y),x)+b*coth(mu*y)*diff(w(x,y),y) = w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y),'build')),output='realtime'));
 

\[w \left ( x,y \right ) ={{\it \_C1}\,{\it \_C2} \left ( {\frac {\sinh \left ( x\lambda \right ) +\cosh \left ( x\lambda \right ) }{\cosh \left ( x\lambda \right ) }} \right ) ^{-{\frac {{\it \_c}_{{1}}}{2\,\lambda }}} \left ( {\frac {\sinh \left ( x\lambda \right ) }{\cosh \left ( x\lambda \right ) }} \right ) ^{{\frac {{\it \_c}_{{1}}}{\lambda }}} \left ( {\frac {\sinh \left ( x\lambda \right ) -\cosh \left ( x\lambda \right ) }{\cosh \left ( x\lambda \right ) }} \right ) ^{-{\frac {{\it \_c}_{{1}}}{2\,\lambda }}} \left ( {\frac {\cosh \left ( \mu \,y \right ) }{\sinh \left ( \mu \,y \right ) }} \right ) ^{{\frac {1}{\mu \,b}}} \left ( {\frac {\cosh \left ( \mu \,y \right ) -\sinh \left ( \mu \,y \right ) }{\sinh \left ( \mu \,y \right ) }} \right ) ^{{\frac {a{\it \_c}_{{1}}}{2\,\mu \,b}}} \left ( {\frac {\cosh \left ( \mu \,y \right ) -\sinh \left ( \mu \,y \right ) }{\sinh \left ( \mu \,y \right ) }} \right ) ^{-{\frac {1}{2\,\mu \,b}}} \left ( {\frac {\cosh \left ( \mu \,y \right ) +\sinh \left ( \mu \,y \right ) }{\sinh \left ( \mu \,y \right ) }} \right ) ^{{\frac {a{\it \_c}_{{1}}}{2\,\mu \,b}}} \left ( {\frac {\cosh \left ( \mu \,y \right ) +\sinh \left ( \mu \,y \right ) }{\sinh \left ( \mu \,y \right ) }} \right ) ^{-{\frac {1}{2\,\mu \,b}}} \left ( \left ( {\frac {\cosh \left ( \mu \,y \right ) }{\sinh \left ( \mu \,y \right ) }} \right ) ^{{\frac {a{\it \_c}_{{1}}}{\mu \,b}}} \right ) ^{-1}}\]

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