I want to solve $y''+y =\begin{cases} t & t\leq 1 \\ 0 & t>1 \end{cases}$ where $y(0)=0$ and $y'(\pi)=1$ using the Laplace transform. I think I can write the piecewise function as $f(t)=t(u(t)-u(t-1))$ but I'm unsure about what to do with the boundaries, since
$\mathcal{L}\{y''\} = s^2\mathcal{L}\{y\}-sy(0)-y'(0)$ and $\mathcal{L}\{y'\} = s\mathcal{L}\{y\}-y(0)$.
I thought about defining a new variable say $t=x+\pi$, such that when $t=\pi$, $x=0$ but when $t=0$, $x=-\pi$ which creates the same problem