Laplace Transforms · Superchallenge problem

A continuous function $y(t)$, $t \ge 0$, of exponential order has Laplace transform $Y(s)$ satisfying $Y(s) - Y(s+1) = \dfrac{1}{s^2}$ for every $s > 0$. Find $Y(2) = \displaystyle\int_0^\infty e^{-2t}y(t)\,dt$. (You may use $\displaystyle\sum_{n\ge1}\dfrac{1}{n^2} = \dfrac{\pi^2}{6}$.)