all!
For my class I have to expand the following equation
$y''(x)=4(\omega^2+q(x))y(x)$
in Fourier coefficients
$y(x)=\frac{1}{2}y_0 + \sum^\infty_{n=1}y_n \cos(2nx)$
$q(x)=2\sum^\infty_{n=1}t_n \cos(2nx)$
The sheet asks me to find the relation
$(\delta_{n0}+\frac{t_n}{\omega^2+n^2})y_0+\sum^\infty_{n=1}(\delta_{nm}+\frac{t_{|n+m|}+t_{|n-m|}}{\omega^2+n^2})y_m = 0$
for all n
I have only gotten as far as
$(\frac{1}{2}\delta_{n0}+\frac{t_n \cos(2nx)}{\omega^2+n^2})y_0+\sum^\infty_{n=1}[\delta_{nm}\cos(2nx)+\frac{t_n}{\omega^2+n^2}\cos(2x(m-n))+\cos(2x(m+n))]y_m = 0$
By substituting, term-wise differentiation and a cosine identity.
I've tried sorting the remaining problems out by inserting specific values for $x$ (such as $0, \frac{\pi}{4}$) and summing the resulting equations, but haven't succeeded.
My biggest concern is how to get from
$\frac{t_n}{\omega^2+n^2}\cos(2x(m-n))+\cos(2x(m+n))$
to
$\frac{t_{|n+m|}+t_{|n-m|}}{\omega^2+n^2}$
They already have kind of a similar structure. Maybe you can point me in the right direction.
Thanks!