$$x(t) = 4y''(t)+4y'(t)+17y(t)$$
where,
$$x(t) = \sum^\infty_{k=1}\left((-1)^{k}\left(-\frac{2\cdot}{k}\right)\sin kt\right) = \sum^\infty_{k=-\infty}\frac{j(-1)^k}{k}e^{jkt},\quad k\not=0$$
and,
$$y(t) = \sum^\infty_{k=-\infty}c_ke^{jk\omega_0t}$$
- $x(t)$ is an input signal, and $y(t)$ is an output signal.
- Objective: Find an expression for $c_k$
Without preparing anything about differential equations in my textbook about transformations, this appears. I assume it's possible to figure this out, but I'm stuck after trying for more than 2 hours.
Does the fact that $x(t)$ is an input signal and that $y(t)$ is an output signal even matter in finding an expression for $c_k$?
Hint: what is the derivative (and the second derivative) of y (t)? If you got that, just plug everything together with the formula for x (t) into the big formula. Now, use the fact that the complex exponentials are independent to get rid of the sums. Finally, you can solve for each $c_k$ individually...
The note with x being input and y being output doesn't change anything. You could change the roles or declare both to be outputs, does change nothing...