Solve this ODE

1. Jan 17, 2004

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...

$$\ddot{f} + b tan(b t) \dot{f} - a^2 cos^2(b t) f = 0$$

Last edited by a moderator: Jan 17, 2004
2. Jan 17, 2004

dhris

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

$$\tau = \sin bt$$

dhris

Last edited: Jan 17, 2004
3. Jan 18, 2004

MathNerd

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

$$\tau = cos(bt)$$

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

4. Jan 19, 2004

dhris

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