Output of a system given it's transfer function and input (beginner)

I have an exercise that gives me the transfer function of a system

$$H(s) = \frac{3s^2+27}{s^4+8s^3 + 16s^2} $$

and an input

$$x(t) = \frac13 cos(3t) $$

An ask's what is the output

I don't whant the answer but the steps to calculate it.

Where to start? I need to find the inverse or there's a shorcut?

Answer

For an LTI system, output $y(t)$ is given by $$y(t) = h(t)\otimes x(t)$$ Where $x(t)$ is input and $h(t)$ is impulse response of the system. The operator $\otimes$ represents convolution.

Convolution operation is mapped into multiplication in Laplace domain. ie, $$Y(s) = H(s)\times X(s)$$ Where, $Y(s)$, $H(s)$ and $X(s)$ are the Laplace transform of $y(t)$, $h(t)$ and $x(t)$ respectively.

You can use any one of the above equation to solve your problem.

Two ways:

1. Find $h(t)$ from $H(s)$ and use first equation. OR


2. Find $X(s)$ from $x(t)$, use second equation to find $Y(s)$ and then find $y(t)$ from it.

I prefer second method.