Thank you Alan 
So I gave it a try (not really involving a(t) yet)
and this is what I have
$$\begin{array}{rll} \frac{dy}{dt} = y-t^2\\
\frac{dy}{dt} - y = -t^2\\
e^{-t}\frac{dy}{dt}-e^{-t}y = -e^{-t}t^2\\
\frac{d}{dt}(e^{-t}y) = -e^{-t}t^2\\
\end{array}$$
$$\begin{array}{rll} e^{-t}y = e^{-t}(t^2+2t+2)+c\\
\mbox{Since }y(0)=y_0\\
e^0y_0 = e^0(0^2-2*0+2)+c\\
c = y_0-2\\
y(t) = t^2+2t+2+y_0-2\\
y(t) = t^2+2t+y_0
\end{array}$$
Does that make sense?
edit: (I also did it for a(t) = t, but I just found my mistake)
