Duhamel's Principle

$$y^{\prime \prime }+a{y}^{\prime }+by=f(t)$$

Find homogenous solution

$$y_h^{\prime \prime }+a{y}_h^{\prime }+b{y}_h=0$$

Solve for $h(t)$, using the Laplace Transform

$$h^{\prime \prime }+a{h}^{\prime }+bh=\delta (t)$$ $$y_p(t)=h\ast f$$

where $h\ast f$ is the Convolution of $h$ and $f$

$$y(t)=y_h(t)+y_p(t)$$