Proof to n-th order inhomogenous differential equation

Question

Let be $ I \subset \mathbb{R} $ an intervall and $ s \in C^{ ( \infty ) }( I) $

How can I show that every solution $ y \in C^{ (n)} (I) $ of $$ y^{ (n)} + \sum_{j=0}^{n-1} a_jy^{(j)} = s(x) $$

( $ a_0,...,a_{n-1} \in \mathbb{R} $ constant ) is in $ C^{( \infty )} (I) $ ?

Answer 1

If by absurd the solution $y$ were not $C^{\infty}$ it means that the left hand side of your equation is not $C^{\infty}$, but the r.h.s is $C^{\infty}$, and this means that the $y$ is not a solution, this a contraddiction. If $y$ is a solution it has to be $C^{\infty}$, if it's not $C^{\infty}$ it can't be a solution.