Differential equation

Question

What method should I follow if I want to solve the equation

$u''-u=\delta_0+\delta_1$ in $\mathcal{D}'(\mathbb{R})$ ??

Thanks in advance!

Answer 1

Try to take the solutions in the form $g(t)h(t)$, where $g(t)\in\mathcal C^2(\Bbb R)$ and $$h(t)=\begin{cases}1,&t\in[0,1],\\0,&\mathrm{otherwise.}\end{cases} $$ Write the equation on $g$ (it will involve a couple of integration by parts), then solve for $g$. Next step will be to find all solutions of the homogenous equation (i.e. with zero right part). This shouldn't be too difficult, but if you need further details, just ask=)