Q:

Where can I use the initial conditions to this general differential equation?

Find in the form of an integral, the solution of the equation
  $$\alpha\frac{dy}{dt}+y=f(t)$$ that satisfies $y\rightarrow 0$ as
  $t\rightarrow -\infty$. Here $f(t)$ is a general function and 
  $\alpha$ is a positive constant.

I'm just a little confused as I get the solution
$y=\frac1{\alpha}e^{-\frac1{\alpha}t}\int e^{\frac1{\alpha}t}f(t) \space dt$
and I don't know how to use the initial conditions- where can I put them in?
Thank you

A:

Note that we have
$$\frac{d}{dt}\left(e^{t/\alpha}y(t)\right)=\frac1\alpha e^{t/\alpha}f(t) \tag 1$$
with $\lim_{t\to -\infty}y(t)=0$.  Integrating $(1)$ from $-\infty$ to $t$ yields
$$y(t)=\frac1\alpha e^{-t/\alpha}\int_{-\infty}^t e^{t'/\alpha}f(t')\,dt'$$

