Consider the differential equation $y''' + 5y'' + 8y' + 6y = f(x)$. For which following input signals $f(x)$ is the response resonant? Choose all those that apply.
$e^x, e^{-x}, e^{3x}, e^{-3x}, e^x\cos(x), e^{-x}\cos(x), e^x\sin(x), e^{-x}\sin(x), e^{-x}\sin(3x), e^{-3x}\sin(3x)$
Using the ERF formula, $z_p = e^{rt}/P(r)$ where $P(r) = r^3 + 5r^2 + 8r + 6$ and the system is in resonance if $P(r) = 0$. Clearly, $r=-3$ works and hence one of the answers if $e^{-3x}$. However, I'm not sure how to proceed with the choices that multiplies $e^{\omega x}$ with sinusoids.
Am I right in thinking that their P(r) equation stays the same and the input (r) can just be the coefficient of the exponent?
Solutions of $r^3 + 5r^2 + 8r + 6=0$ is $r=-3$, $r=-1\pm i$.
Then
$f(x)=e^{-x}\sin x$ or $f(x)=e^{-x}\cos x$