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'));
 

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

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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'));
 

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

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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'));
 

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

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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'));
 

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

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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'));
 

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

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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'));
 

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

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