Does the equation $x''+\lambda x' - x = \phi$, where $\lambda\in\mathbb{R}$ and $\phi\in\mathcal{D}(\mathbb{R}^N)$ is a given arbitrary test function, always have a solution in a distributional sense?
I think that the answer is affirmative. And the reason is because the Malgrange-Ehrenpreis theorem guarantees the existence of the fundamental solution, let's say $T$. So then, the distributional solution of $x''+\lambda x' - x = \phi$ is $T\ast \phi$. Is this reasoning correct?
Thanks in advance.