6.2.16 6.2

6.2.16.1 [653] problem number 1

problem number 653

Added January 14, 2019.

Problem 2.6.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x +\left ( a \cos ^k(\lambda x)+b \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*Cos[lambda*x]^k + b)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {a \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{k+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\cos ^2(\lambda x)\right )}{k \lambda +\lambda }-b x+y\right )\right \}\right \}\]

Maple ✓

restart; 
pde := diff(w(x,y),x)+(a*cos(lambda*x)^k+b)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (-b x +y -\left (\int a \left (\cos ^{k}\left (\lambda x \right )\right )d x \right )\right )\] Contains unresolved integral

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6.2.16.2 [654] problem number 2

problem number 654

Added January 14, 2019.

Problem 2.6.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x +\left ( a \cos ^k(\lambda y)+b \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*Cos[lambda*y]^k + b)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde := diff(w(x,y),x)+(a*cos(lambda*y)^k+b)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (x -\left (\int \frac {1}{a \left (\cos ^{k}\left (\lambda y \right )\right )+b}d y \right )\right )\]

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6.2.16.3 [655] problem number 3

problem number 655

Added January 14, 2019.

Problem 2.6.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x +a \cos ^k(\lambda x) \cos ^n(\mu y) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + a*Cos[lambda*y]^k*Cos[mu*y]^n*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\int _1^y\cos ^{-k}(\lambda K[1]) \cos ^{-n}(\mu K[1])dK[1]-a x\right )\right \}\right \}\]

Maple ✓

restart; 
pde := diff(w(x,y),x)+a*cos(lambda*y)^k*cos(mu*y)^n*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {a x -\left (\int \left (\cos ^{-n}\left (\mu y \right )\right ) \left (\cos ^{-k}\left (\lambda y \right )\right )d y \right )}{a}\right )\]

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6.2.16.4 [656] problem number 4

problem number 656

Added January 14, 2019.

Problem 2.6.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x +a \cos ^k(x+\lambda y) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + a*Cos[x + lambda*y]^k*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := diff(w(x,y),x)+a*cos(x+lambda*y)^k*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (-\lambda \left (\int _{}^{\frac {\lambda y +x}{\lambda }}\frac {1}{a \lambda \left (\cos ^{k}\left (\mathit {\_a} \lambda \right )\right )+1}d\mathit {\_a} \right )+x \right )\]

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6.2.16.5 [657] problem number 5

problem number 657

Added January 14, 2019.

Problem 2.6.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x +\left ( y^2-a^2 + a \lambda \cos (\lambda x) + a^2 \cos ^2(\lambda x) \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 - a^2 + a*lambda*Cos[lambda*x] + a^2*Cos[lambda*x]^2)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := diff(w(x,y),x)+( y^2-a^2 + a *lambda*cos(lambda*x) + a^2*cos(lambda*x)^2)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (-\frac {2 \sqrt {2 \cos \left (\lambda x \right )+2}\, \left (\frac {\lambda \HeunCPrime \left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \sin \left (\lambda x \right )}{2}+\left (a \sin \left (\lambda x \right )-y \right ) \HeunC \left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )\right )}{2 \left (\cos \left (\lambda x \right )+1\right ) \lambda \HeunCPrime \left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \sin \left (\lambda x \right )+\left (-4 \left (\cos \left (\lambda x \right )+1\right ) y +\left (4 a \cos \left (\lambda x \right )+4 a +2 \lambda \right ) \sin \left (\lambda x \right )\right ) \HeunC \left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )}\right )\]

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6.2.16.6 [658] problem number 6

problem number 658

Added January 14, 2019.

Problem 2.6.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x +\left ( \lambda \cos (\lambda x) y^2 + \lambda \cos ^3(\lambda x) \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*Cos[lambda*x]*y^2 + lambda*Cos[lambda*x]^3)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := diff(w(x,y),x)+(lambda*cos(lambda*x)*y^2 + lambda*cos(lambda*x)^3)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {-\left (y -\sin \left (\lambda x \right )\right ) M\left (1, \frac {3}{2}, -\left (\sin ^{2}\left (\lambda x \right )\right )\right ) \sin \left (\lambda x \right )-1}{\left (y -\sin \left (\lambda x \right )\right ) U\left (1, \frac {3}{2}, -\left (\sin ^{2}\left (\lambda x \right )\right )\right ) \sin \left (\lambda x \right )-2}\right )\]

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6.2.16.7 [659] problem number 7

problem number 659

Added January 14, 2019.

Problem 2.6.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ 2 w_x +\left ( (\lambda +a+a \cos (\lambda x)) y^2 + \lambda - a + a \cos (\lambda x) \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  2*D[w[x, y], x] + ((lambda + a + a*Cos[lambda*x])*y^2 + lambda - a + a*Cos[lambda*x])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := 2*diff(w(x,y),x)+ ((lambda+a+a*cos(lambda*x))*y^2 +lambda - a + a *cos(lambda*x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (y \cos \left (\lambda x \right )+y -\sin \left (\lambda x \right )\right ) \sqrt {\cos \left (\lambda x \right )+1}\, \sqrt {\cos \left (\lambda x \right )-1}\, {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }}}{\left (\left (y \cos \left (\lambda x \right )+y -\sin \left (\lambda x \right )\right ) \sqrt {\cos \left (\lambda x \right )+1}\, \sqrt {\cos \left (\lambda x \right )-1}\, \left (\int \frac {\left (a \cos \left (\lambda x \right )+a +\lambda \right ) {\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{\lambda }} \sin \left (\lambda x \right )}{\sqrt {\cos \left (\lambda x \right )-1}\, \left (\cos \left (\lambda x \right )+1\right )^{\frac {3}{2}}}d x \right ) {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }}+2 \sin \left (\lambda x \right )\right ) \lambda }\right )\]

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6.2.16.8 [660] problem number 8

problem number 660

Added January 14, 2019.

Problem 2.6.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x +\left ( (\lambda +a \cos ^2(\lambda x)) y^2 + \lambda - a + a \cos ^2(\lambda x) \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + ((lambda + a*Cos[lambda*x]^2)*y^2 + lambda - a + a*Cos[lambda*x]^2)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := diff(w(x,y),x)+ ((lambda+a*cos(lambda*x)^2)*y^2 + lambda - a + a*cos(lambda*x)^2)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {2 \left (\left (a \left (\cos ^{2}\left (\lambda x \right )\right )+\lambda \right ) y \left (\cos ^{4}\left (\lambda x \right )\right )-\frac {\left (\cos \left (2 \lambda x \right )+1\right ) \left (a \cos \left (2 \lambda x \right )+a +2 \lambda \right ) \sin \left (2 \lambda x \right )}{8}\right ) \sqrt {\cos \left (2 \lambda x \right )-1}\, {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }}}{\left (4 \left (\left (a \left (\cos ^{2}\left (\lambda x \right )\right )+\lambda \right ) y \left (\cos ^{4}\left (\lambda x \right )\right )-\frac {\left (\cos \left (2 \lambda x \right )+1\right ) \left (a \cos \left (2 \lambda x \right )+a +2 \lambda \right ) \sin \left (2 \lambda x \right )}{8}\right ) \sqrt {\cos \left (2 \lambda x \right )-1}\, \left (\int \frac {\left (a \cos \left (2 \lambda x \right )+a +2 \lambda \right ) {\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \sin \left (2 \lambda x \right )}{\sqrt {\cos \left (2 \lambda x \right )-1}\, \left (\cos \left (2 \lambda x \right )+1\right )^{\frac {3}{2}}}d x \right ) {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }}+\sqrt {\cos \left (2 \lambda x \right )+1}\, \left (a \cos \left (2 \lambda x \right )+a +2 \lambda \right ) \sin \left (2 \lambda x \right )\right ) \lambda }\right )\]

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6.2.16.9 [661] problem number 9

problem number 661

Added January 14, 2019.

Problem 2.6.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ (a x^n y^m+b x) w_x +\cos ^k(\lambda y) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n*y^m + b*x)*D[w[x, y], x] + Cos[lambda*y]^k*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  (a*x^n*y^m+b*x)*diff(w(x,y),x)+ cos(lambda*y)^k*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\left (n -1\right ) a \left (\int y^{m} \left (\cos ^{-k}\left (\lambda y \right )\right ) {\mathrm e}^{\left (n -1\right ) b \left (\int \left (\cos ^{-k}\left (\lambda y \right )\right )d y \right )}d y \right )+x^{-n +1} {\mathrm e}^{\left (n -1\right ) b \left (\int \left (\cos ^{-k}\left (\lambda y \right )\right )d y \right )}\right )\]

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6.2.16.10 [662] problem number 10

problem number 662

Added January 14, 2019.

Problem 2.6.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ (a x^n +b x \cos ^m y) w_x +y^k w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n + b*x*Cos[y]^m)*D[w[x, y], x] + y^k*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  (a*x^n+b*x*cos(y)^m)*diff(w(x,y),x)+y^k*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\left (n -1\right ) a \left (\int y^{-k} {\mathrm e}^{\left (n -1\right ) b \left (\int y^{-k} \left (\cos ^{m}y \right )d y \right )}d y \right )+x^{-n +1} {\mathrm e}^{\left (n -1\right ) b \left (\int y^{-k} \left (\cos ^{m}y \right )d y \right )}\right )\]

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6.2.16.11 [663] problem number 11

problem number 663

Added January 14, 2019.

Problem 2.6.2.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ (a x^n +b x \cos ^m y) w_x + \cos ^k(\lambda y) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n + b*x*Cos[y]^m)*D[w[x, y], x] + Cos[lambda*y]^k*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  (a*x^n+b*x*cos(y)^m)*diff(w(x,y),x)+cos(lambda*y)^k*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\left (n -1\right ) a \left (\int \left (\cos ^{-k}\left (\lambda y \right )\right ) {\mathrm e}^{\left (n -1\right ) b \left (\int \left (\cos ^{m}y \right ) \left (\cos ^{-k}\left (\lambda y \right )\right )d y \right )}d y \right )+x^{-n +1} {\mathrm e}^{\left (n -1\right ) b \left (\int \left (\cos ^{m}y \right ) \left (\cos ^{-k}\left (\lambda y \right )\right )d y \right )}\right )\]

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6.2.16.12 [664] problem number 12

problem number 664

Added January 14, 2019.

Problem 2.6.2.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ (a x^n \cos ^m y + b x) w_x + \cos ^k(\lambda y) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n*Cos[y]^m + b*x)*D[w[x, y], x] + Cos[lambda*y]^k*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  (a*x^n*cos(y)^m+b*x)*diff(w(x,y),x)+cos(lambda*y)^k*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\left (n -1\right ) a \left (\int \left (\cos ^{m}y \right ) \left (\cos ^{-k}\left (\lambda y \right )\right ) {\mathrm e}^{\left (n -1\right ) b \left (\int \left (\cos ^{-k}\left (\lambda y \right )\right )d y \right )}d y \right )+x^{-n +1} {\mathrm e}^{\left (n -1\right ) b \left (\int \left (\cos ^{-k}\left (\lambda y \right )\right )d y \right )}\right )\]

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