A question asks us to solve the differential equation
$-u''(x) = \delta(x)$
with boundary conditions
$u(-2) = 0$ and $u(3) = 0$ where $\delta(x)$ is the Dirac delta function. But inside the same question, teacher gives the solution in two pieces as $u = A(x+2)$ for $x\le0$ and $u = B(x-3)$ for $x \ge 0$. I understand when we integrate the delta function twice the result is the ramp function $R(x)$. However elsewhere in his lecture the teacher had given the general solution of that DE as
$u(x) = -R(x) + C + Dx$
So I dont understand how he was able to jump from this solution to the two pieces. Are these the only two pieces possible, using the boundary conditions given, or can there be other solutions?
Full solution is here (section 1.2 answer #2)