Find distributions whose second derivative is Dirac delta

Question

How can I find distributions $T \in \mathcal{D'}(\mathbb{R})$ such that $T'' = \delta _0$ ?

I know that $D^2T(\phi) = T(D^2 \phi)$.

$\phi \in C^{\infty}(\Omega, \mathbb{R}), supp \ \phi$ is compact

How can I use it to solve this equation?

Answer 1

First step: prove that the distribution with zero derivative are constant functions.

Second step: learn to solve the equation $T'=\delta_0$ (easy to solve; if you can't, search math.SE, there's a question about is under the 'distribution-theory' tag).

Third step: learn to solve the differential equation $S'= G$, where $G$ is a piece-wise continuous function.