Let \(\displaystyle{\left(f_{m}\right)_{m\in\mathbb{Z}}}\) be a complex sequence such that
\(\displaystyle{\sum_{m\in\mathbb{Z}}f_m<\infty}\). Consider the continuous function \(\displaystyle{f:\left(\pi,\pi\right]\to \mathbb{C}}\)
defined by \(\displaystyle{f(t)=\sum_{m\in\mathbb{Z}}f_m\,e^{i\,m\,t}}\) and the differential equation
\(\displaystyle{x''+2\,k\,x'+d^2\,x=f}\) , where \(\displaystyle{k\,,d}\) are positive constants.
Suppose a solution defined by \(\displaystyle{x(t)=\sum_{m\in\mathbb{Z}}x_m\,e^{i\,m\,t}\,,\pi<t\leq \pi}\)
and find \(\displaystyle{x_m\,,m\in\mathbb{Z}}\).
Note
The complex space \(\displaystyle{\left(C((\pi,\pi]),+\right)}\) has an inner product
\(\displaystyle{\langle{f,g\rangle}=\dfrac{1}{2\,\pi}\,\int_{\pi}^{\pi}f(t)\,\overline{g(t)}\,\mathrm{d}t\,\,\forall\,f\,,g\in C((\pi,\pi])}\).
