Problem $3$

Given the following differential equation, where $x(t)$ is the position and $f(t)$ is a physical force

$\frac{d^{2}x}{d{t}^2}+12\frac{dx}{dt}+32x=f(t)$

Use Laplace transform to find the transfer function $G(s)=\frac{x(s)}{F(s)}.$

Let the input $F(s)=\frac{32}{s},$ find the time response of $x(t)$ in relation to the input $f(t).$ Hint: Use partial fraction to transform $x(s)$ to $x(t)$

The time response $x(t)$ is composed of two parts, highlight them in your solution.

What is the assumption that you made regarding the initial conditions and what is its physical meaning?