What am I doing wrong in the Fourier expansion?

[Ne0]

Ok we are given the ODE
<br /> {y}^{\prime\prime}(t) + \omega^2{y(t)} = {r(t)} <br />
<br /> r(t) = cos\omega{t} <br />
\omega = 0.5,0.8,1.1,1.5,5.0,10.0 <br />
I know you can use variation of paramaters to solve for it so I start by finding the complementary solution.
<br /> {y}^{\prime\prime}(t) + \omega^2{y(t)} = 0<br />
We know solutions are of the form
<br /> y = \exp{(mt)} <br />
so after taking derivatives and what not we get the fundamental solution
<br /> \cos\omega{t}, \sin\omega{t}<br />
Our complementary solution is
<br /> {y}_{c}=Acos \omega{t} + Bsin \omega{t}<br />
For the particular solution we set
<br /> {y}^{\prime\prime}(t) + \omega^2{y(t)} = cos\omega{t} <br />
We then use a Fourier series to expand
<br /> cos\omega{t}<br />
Then proceed to solve for it but the problem I'm having is that I'm getting the Fourier series to be zero which is strange. I know that there will be no
<br /> {b}_{n}<br />
term since cos is even but its still werid why I'm getting zero for
<br /> {a}_{0}, {a}_{n}<br />
Any help would be appreciated.

 

[Ne0]

I swear this latex thing I can't figure it out.

 

[Ne0]

If you guys are stuck the answer in the book is:
<br /> {y} = {c}_{1}\cos\omega{t} + {c}_{2}\sin\omega{t} + {A}(\omega)\cos\omega{t}<br />

<br /> {A}(\omega) = \frac{1}{\omega^2 - 1} {\leq} 0<br />
if \omega^2 {\leq} 1

<br /> {A}(\omega) = \frac{1}{\omega^2 - 1} {\geq} 0<br />
if \omega^2 {\geq} 1

Since there was not only greater then and less then I had to use less then or equal and greater then or equal

 

[Ne0]

Can anyone help please with what I am doing wrong in the Fourier expansion?