I am working on solving a PDE with the wave equation and d'Alembert's formula, where there is a boundary condition. I am supposed to use an odd reflection of the wave equation. The IVP is
$$u_{tt}-u_{xx}=0, x>0, t>0$$ $$u(0,t)=0, t>0$$ $$u(x, 0)=0, x>0$$ $$u_t(x, 0)=p(x)$$, where $p(x)= \left\{ \begin{array}{c} x, 0\le x\le1 \\ 0, \text{otherwise} \\ \end{array} \right. $
I think there are 2 cases: $x \ge t\gt 0$ and $t\ge x\gt 0$, and I know that the first case could be solved with the regular d'Alembert formula, and the second case needs the odd reflection, making it $$u(x,t)=\frac12 \int_{t-x}^{x+t} p(s)ds$$ because $u(x,0)=0$. I just do not know where to go from here. Any help would be appreciated!!!