6.3.20 6.5

6.3.20.1 [957] Problem 1

problem number 957

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'));
 

\[w \left ( x,y \right ) ={\frac {kx}{a}}+{\frac {1}{b\mu \,a\lambda } \left ( {\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) b\mu \,a\lambda +c\sin \left ( \mu \,y \right ) a\lambda -\cos \left ( \lambda \,x \right ) b\mu \right ) }\]

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6.3.20.2 [958] Problem 2

problem number 958

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'));
 

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

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6.3.20.3 [959] Problem 3

problem number 959

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'));
 

\[w \left ( x,y \right ) ={\frac {cx}{a}}-1/2\,{\frac {\cos \left ( \lambda \,x-\mu \,y \right ) \left ( a\lambda +b\mu \right ) +\cos \left ( \lambda \,x+\mu \,y \right ) \left ( a\lambda -b\mu \right ) }{ \left ( a\lambda -b\mu \right ) \left ( a\lambda +b\mu \right ) }}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \]

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6.3.20.4 [960] Problem 4

problem number 960

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'));
 

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

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6.3.20.5 [961] Problem 5

problem number 961

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]];
 

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

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'));
 

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

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6.3.20.6 [962] Problem 6

problem number 962

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'));
 

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

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