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'));
 

\[w \left (x , y\right ) = \frac {x k +\frac {f_{1} \left (\frac {y a -b x}{a}\right ) \mu b a \lambda +c a \sin \left (\mu y \right ) \lambda -\cos \left (\lambda x \right ) \mu b}{b \lambda \mu }}{a}\]

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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]];
 

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

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'));
 

\[w \left (x , y\right ) = \frac {x k}{a}-\frac {c \cos \left (\mu y \right )}{\mu b}+\frac {\ln \left (\sec \left (\lambda x \right )^{2}\right )}{2 \lambda a}+f_{1} \left (\frac {y a -b x}{a}\right )\]

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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]];
 

\[\left \{\left \{w(x,y)\to \frac {e^{-i (\lambda x+\mu y)} \left (4 a \left (a^2 \lambda ^2-b^2 \mu ^2\right ) e^{i (\lambda x+\mu y)} c_1\left (y-\frac {b x}{a}\right )+a^2 \lambda \left (4 c \lambda x e^{i (\lambda x+\mu y)}-e^{2 i (\lambda x+\mu y)}-e^{2 i \lambda x}-e^{2 i \mu y}-1\right )+a b \mu \left (-1+e^{2 i \lambda x}\right ) \left (-1+e^{2 i \mu y}\right )-4 b^2 c \mu ^2 x e^{i (\lambda x+\mu y)}\right )}{4 a (a \lambda -b \mu ) (a \lambda +b \mu )}\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = \frac {x c}{a}-\frac {\cos \left (\lambda x -\mu y \right )}{2 \lambda a -2 \mu b}-\frac {\cos \left (\lambda x +\mu y \right )}{2 \lambda a +2 \mu b}+f_{1} \left (\frac {y a -b x}{a}\right )\]

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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]];
 

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

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'));
 

\[w \left (x , y\right ) = \frac {f_{1} \left (\frac {x b \mu +a \ln \left (\csc \left (\mu y \right )+\cot \left (\mu y \right )\right )}{b \mu }\right ) b \mu \lambda a -c \ln \left (\csc \left (\mu y \right )+\cot \left (\mu y \right )\right ) \lambda a +\sin \left (\lambda x \right ) b \mu }{b \mu \lambda a}\]

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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]];
 

\[\left \{\left \{w(x,y)\to \int _1^x\frac {c+\sin (\lambda K[1])}{a}dK[1]+c_1\left (\frac {\log (\sin (\mu y))}{\mu }-\frac {b x}{a}\right )\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = \frac {f_{1} \left (\frac {-x b \mu +\ln \left (\operatorname {csgn}\left (\sec \left (\mu y \right )\right ) \sin \left (\mu y \right )\right ) a}{b \mu }\right ) a \lambda +x c \lambda -\cos \left (\lambda x \right )}{a \lambda }\]

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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'));
 

\[w \left (x , y\right ) = \frac {2 f_{1} \left (\frac {-x b \mu +\ln \left (\operatorname {csgn}\left (\sec \left (\mu y \right )\right ) \sin \left (\mu y \right )\right ) a}{b \mu }\right ) a \lambda +2 x c \lambda -\ln \left (\csc \left (\lambda x \right )^{2}\right )}{2 a \lambda }\]

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