6.4.19 6.5

6.4.19.1 [1134] Problem 1

problem number 1134

Added March 9, 2019.

Problem Chapter 4.6.5.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

____________________________________________________________________________________

6.4.19.2 [1135] Problem 2

problem number 1135

Added March 9, 2019.

Problem Chapter 4.6.5.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

____________________________________________________________________________________

6.4.19.3 [1136] Problem 3

problem number 1136

Added March 9, 2019.

Problem Chapter 4.6.5.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

____________________________________________________________________________________

6.4.19.4 [1137] Problem 4

problem number 1137

Added March 9, 2019.

Problem Chapter 4.6.5.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\ln \left (\mathrm {csgn}\left (\frac {1}{\cos \left (\mu y \right )}\right ) {\mathrm e}^{-a \mu x} \sin \left (\mu y \right )\right )}{a \mu }\right ) {\mathrm e}^{-\frac {b \cos \left (\lambda x \right )}{\lambda }}\]

____________________________________________________________________________________

6.4.19.5 [1138] Problem 5

problem number 1138

Added March 9, 2019.

Problem Chapter 4.6.5.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {-a \ln \left (-\cot \left (\lambda x \right )+\csc \left (\lambda x \right )\right )+\lambda y}{\lambda }\right ) {\mathrm e}^{\frac {b \sin \left (\mu y \right )}{a \mu }}\]

____________________________________________________________________________________

6.4.19.6 [1139] Problem 6

problem number 1139

Added March 9, 2019.

Problem Chapter 4.6.5.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

____________________________________________________________________________________