Solutions to linear differential equations can be written using convolutions as
$$y = y_{IVP} + (h(t) * f(t))$$
• $y_{IVP}$ is the solution to the associated homogeneous differential equation with the given initial values (ignore the forcing function, keep initial values).
• $h(t)$ is the impulse response (ignore the initial values and forcing function).
• $f(t)$ is the forcing function. (ignore the initial values and differential equation).
Use the form above to write the solution to the differential equation
$$y'' + 3y' = -3te^{-3t}$$ with $$y(0) = 5, y'(0) = -6$$
$$y = \boxed{y_{IVP}} + (\boxed{h(t)} * \boxed{f(t)})$$
