Solving ODE involving piece-wise function This question may sound very trivial but I really can't figure out how to solve this. It has been a while since I solved ODEs and I need to solve an equation similar to one given below for some application. Basically it's an ODE with a piece wise definition. $$ \frac{d^{2}u}{dx^{2}}+Q(x)=0 \quad \text{For} \; 0<=x<=1\\\ Q(x) = \begin{cases} 0, & 0<=x<=0.5 \\\ 20 – 40x, & 0.5<x<=0.75 \\\ 40x – 40, & x>0.75 \\\ \end{cases}\\\ \text{with boundary conditions}\\\ u(0)=0\\\ u(1)=0 $$ How do I go about solving it? Of course I can't just integrate it. I don't need someone to solve entire problem for me. Just shepherding me in right direction should be enough.

Solve the equation three times: one in each of the regions singled out by the definition of $Q(x)$. There will be six constants of integration (three second-order differential equations) and you will want to enforce continuity of $u(x)$ and $u'(x)$, as well as the given boundary conditions.