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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {1}{2} \left (\frac {c e^{-\lambda x} \left (e^{2 \lambda x}+1\right )}{a \lambda }+\frac {k e^{-\mu y} \left (e^{2 \mu y}-1\right )}{b \mu }\right )\right )\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {k a \sinh \left (\mu y \right ) \lambda +c \cosh \left (\lambda x \right ) \mu b}{a \lambda \mu b}}\]

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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]];
 

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

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'));
 

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

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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]];
 

\begin{align*}& \left \{w(x,y)\to c_1\left (-\frac {a \mu x+\text {arctanh}(\cosh (\mu y))}{\mu }\right ) \exp \left (\int _1^xb \cosh (\lambda K[1])dK[1]\right )\right \}\\& \left \{w(x,y)\to c_1\left (-\frac {a \mu x+\text {arctanh}(\cosh (\mu y))}{\mu }\right ) \exp \left (\int _1^xb \cosh (\lambda K[2])dK[2]\right )\right \}\\\end{align*}

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'));
 

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

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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]];
 

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

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'));
 

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

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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]];
 

\[\left \{\left \{w(x,y)\to e^{-\frac {\text {arctanh}(\cosh (\lambda x))}{a \lambda }} c_1\left (\frac {b \text {arctanh}(\cosh (\lambda x))}{a \lambda }-\frac {\cot ^{-1}(\sinh (\mu y))}{\mu }\right )\right \}\right \}\]

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'));
 

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

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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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-b \mu \ln \left ({\mathrm e}^{2 \lambda x}-1\right )+\ln \left (-\cosh \left (\mu y \right )\right ) a \lambda +b \mu \left (\lambda x +\ln \left (2\right )\right )}{\lambda b \mu }\right ) \sinh \left (\lambda x \right )^{\frac {1}{a \lambda }}\]

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