Equations with discontinuous coefficients

[jumpingbean35]

This problem has me confused near the end:

Solve the initial value problem
y'+2y=g(t), y(0)=0
where
g(t)= 1, 0 ≤ t ≤ 1
g(t) = 0, t>1

So I solved the equation for each value of g(t) and got
y=1/2(1-e^-2t) where g(t)=1 and
y=e^(c-2t) where g(t)=0

I don't know how to solve for c in the second problem, since the initial value occurs where g(t)=1, not 0. The answer given in the book for the second half of the equation is y=1/2(e^2-1)e^-2t, but I have pretty much no idea how to proceed in order to arrive there
Does anyone have any hints/thoughts/suggestions?

Thanks!