Find the coefficient of this differential equation?

Question

$y(t) = x''(t) + c_1x'(t) + c_0x(t)$

Determine the coefficients $c_1$ And $c_0$ so that the output signal become $y(t) = 0$ when the input signal is $x(t) = sin(w_0t)$ for some given constant $w_0$.

Answer 1

$$y(t) = x''(t) + c_1x'(t) + c_0x(t)$$ If $x(t)=\sin(\omega t)$ $$y(t) = -\omega^2\sin(t) +c_1\omega \cos(t) + c_0\sin(t)$$ $$y(t) = (c_0-\omega^2)\sin(t) +c_1\omega \cos(t) $$ Obviously $y(t)=0$ implies $$c_0=\omega^2 \text{ and } c_1=0$$