7.4.12 4.5

7.4.12.1 [1098] Problem 1

problem number 1098

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'));
 

\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {a k \lambda \sinh \left (\mu y \right )+b c \mu \cosh \left (\lambda x \right )}{a b \lambda \mu }}\]

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7.4.12.2 [1099] Problem 2

problem number 1099

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'));
 

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

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7.4.12.3 [1100] Problem 3

problem number 1100

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'));
 

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

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7.4.12.4 [1101] Problem 4

problem number 1101

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'));
 

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

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7.4.12.5 [1102] Problem 5

problem number 1102

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'));
 

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

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7.4.12.6 [1103] Problem 6

problem number 1103

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'));
 

\[w \left (x , y\right ) = \left (\tanh \left (\lambda x \right )-1\right )^{-\frac {1}{2 a \lambda }} \left (\tanh \left (\lambda x \right )+1\right )^{-\frac {1}{2 a \lambda }} \left (\tanh ^{\frac {1}{a \lambda }}\left (\lambda x \right )\right ) \textit {\_F1} \left (\frac {a \lambda \ln \left (\frac {\sqrt {\left (\coth ^{2}\left (\mu y \right )-1\right ) \left (-\frac {1}{\left ({\mathrm e}^{2 \lambda x}-1\right )^{2}}\right )^{\frac {b \mu }{a \lambda }}}\, \coth \left (\mu y \right )}{\coth ^{2}\left (\mu y \right )-1}\right )+\left (\lambda x +\ln \left (2\right )\right ) b \mu }{b \lambda \mu }\right )\]

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