Thread: How to solve this ODE?..

1. How to solve this ODE?..

I'm reading a book on asymptotic methods and I encountered the following ODE, do you have any clue how to solve this? Thanks.

$\displaystyle y''+y=\frac{\cos(x)}{5+8\cos(x)\sin(x)}$

2. Re: How to solve this ODE?..

Hey Mengqi.

You should try using the complem entary and particular solution.

On the right hand side you have a 2nd order linear DE which can be solved through a standard technique. On the Right hand side you have a particular solution which can be solved using a number of techniques including operator methods (using the d/dx = D as an operator).

These techniques should be taught in any standard differential equations courses.

3. Re: How to solve this ODE?..

Thanks Chiro,

I see. But when I tried to solve it in this way: I know the left hand side will give me the solution like $\displaystyle y=A\cos(x)+B\sin(x)$ (with right hand side being set to zero temporally), then I assume $\displaystyle y_{particular}=C(x)(A\cos(x)+B\sin(x))$ and plug it in, but this way seems not to work. In the end, I got sth. like $\displaystyle [C''(x)(A\cos x+B\sin x)-2C'(x)(A\cos x+B\sin x )](5+8\cos x\sin x)=\cos x$. How to proceed from here? I didn't reduce the problem to a simple one, C still appears with a second order derivative. Do you know any other techniques? Thanks

4. Re: How to solve this ODE?..

And it seems not trivial because I just used Mathematica to see the final result, which is attached. I think it involves some advanced technique? It's great if any of you could solve it. I am curious to see the derivation. Thanks.