Solving the ODE with Arbitrary Constants: A Search for Analytic Solutions

[MathNerd]

I’ve been trying to find an analytic solution to the following ODE. I haven’t been successful and have come to the conclusion that an analytic solution probably doesn’t exist. I am not totally sure though and would be appreciative if you guys gave it a look.

a & b are arbitrary constants...

[tex]
\ddot{f} + b tan(b t) \dot{f} - a^2 cos^2(b t) f = 0
[/tex]

Thanks in advance...

 

[dhris]

First, write the tan as sin/cos. Then multiply the equation by cos(bt). Then write the equation in terms of the variable:

[tex] \tau = \sin bt [/tex]

dhris

 

[MathNerd]

Originally posted by dhris
First, write the tan as sin/cos. Then multiply the equation by cos(bt). Then write the equation in terms of the variable:

[tex] \tau = \sin bt [/tex]

dhris

Thanks for the hint. I actually found this substitution works best…

[tex] \tau = cos(bt) [/tex]

after making the substitution I solved it via a power series method.

 

[dhris]

Originally posted by MathNerd
Thanks for the hint. I actually found this substitution works best…

[tex] \tau = cos(bt) [/tex]

after making the substitution I solved it via a power series method.

But you can solve it exactly if you use the other one!