6.3.20 6.5

6.3.20.1 [956] Problem 1

problem number 956

Added Feb. 11, 2019.

Problem Chapter 3.6.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = \sin (\lambda x)+c \cos (\mu y)+k \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \frac {k x}{a}+\frac {a b \lambda \mu \mathit {\_F1} \left (\frac {y a -b x}{a}\right )+a c \lambda \sin \left (\mu y \right )-b \mu \cos \left (\lambda x \right )}{a b \lambda \mu }\]

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6.3.20.2 [957] Problem 2

problem number 957

Added Feb. 11, 2019.

Problem Chapter 3.6.5.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = \tan (\lambda x)+c \sin (\mu y)+k \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \frac {k x}{a}+\mathit {\_F1} \left (\frac {y a -b x}{a}\right )-\frac {c \cos \left (\mu y \right )}{b \mu }-\frac {\ln \left (\cos \left (\lambda x \right )\right )}{a \lambda }\]

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6.3.20.3 [958] Problem 3

problem number 958

Added Feb. 11, 2019.

Problem Chapter 3.6.5.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = \sin (\lambda x) \cos (\mu y)+c \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \frac {\left (a^3 \lambda ^2-a b^2 \mu ^2\right ) c_1\left (y-\frac {b x}{a}\right )+a^2 c \lambda ^2 x-a^2 \lambda \cos (\lambda x) \cos (\mu y)-a b \mu \sin (\lambda x) \sin (\mu y)-b^2 c \mu ^2 x}{a (a \lambda -b \mu ) (a \lambda +b \mu )}\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = \frac {c x}{a}+\mathit {\_F1} \left (\frac {y a -b x}{a}\right )-\frac {\left (a \lambda +\mu b \right ) \cos \left (\lambda x -\mu y \right )+\left (a \lambda -\mu b \right ) \cos \left (\lambda x +\mu y \right )}{2 \left (a \lambda -\mu b \right ) \left (a \lambda +\mu b \right )}\]

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6.3.20.4 [959] Problem 4

problem number 959

Added Feb. 11, 2019.

Problem Chapter 3.6.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \sin (\mu y) w_y = \cos (\lambda y)+c \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \frac {a \lambda c_1\left (\frac {\log \left (\tan \left (\frac {\mu y}{2}\right )\right )}{\mu }-\frac {b x}{a}\right )+c \lambda x+\sin (\lambda x)}{a \lambda }\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = \frac {a \lambda \mathit {\_F1} \left (\frac {a \ln \left (\RootOf \left (\mu y -\arctan \left (\frac {2 \mathit {\_Z} \,{\mathrm e}^{\frac {b \mu x}{a}}}{\mathit {\_Z}^{2} {\mathrm e}^{\frac {2 b \mu x}{a}}+1}, -\frac {\mathit {\_Z}^{2} {\mathrm e}^{\frac {2 b \mu x}{a}}-1}{\mathit {\_Z}^{2} {\mathrm e}^{\frac {2 b \mu x}{a}}+1}\right )\right )\right )}{b \mu }\right )+c \lambda x +\sin \left (\lambda x \right )}{a \lambda }\]

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6.3.20.5 [960] Problem 5

problem number 960

Added Feb. 11, 2019.

Problem Chapter 3.6.5.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \tan (\mu y) w_y = \sin (\lambda y)+c \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \frac {c \lambda x-\cos (\lambda x)}{a \lambda }+c_1\left (\frac {\log (\sin (\mu y))}{\mu }-\frac {b x}{a}\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = \frac {a \lambda \mathit {\_F1} \left (\frac {-b \mu x +a \ln \left (\frac {\tan \left (\mu y \right )}{\sqrt {\tan ^{2}\left (\mu y \right )+1}}\right )}{b \mu }\right )+c \lambda x -\cos \left (\lambda x \right )}{a \lambda }\]

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6.3.20.6 [961] Problem 6

problem number 961

Added Feb. 11, 2019.

Problem Chapter 3.6.5.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \tan (\mu y) w_y = \cot (\lambda y)+c \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \frac {a \lambda c_1\left (\frac {\log (\sin (\mu y))}{\mu }-\frac {b x}{a}\right )+c \lambda x+\log (\sin (\lambda x))}{a \lambda }\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = \frac {2 a \lambda \mathit {\_F1} \left (\frac {-b \mu x +a \ln \left (\frac {\tan \left (\mu y \right )}{\sqrt {\tan ^{2}\left (\mu y \right )+1}}\right )}{b \mu }\right )+2 c \lambda x -\ln \left (\cot ^{2}\left (\lambda x \right )+1\right )}{2 a \lambda }\]

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