Laplace II: Steps, Impulses & Convolution · Superchallenge problem

A continuous function $y$ on $t\ge0$, of exponential order, satisfies the nonlinear convolution equation $y'(t)=\displaystyle\int_0^t y(\tau)\,y(t-\tau)\,d\tau$ with $y(0)=1$. Use the Laplace transform to find $\displaystyle\int_0^\infty e^{-3t}\,y(t)\,dt$. Give an exact answer.