 Compact support solvability for a constant-coefficient ordinary differential equation

Use $D=-i{\partial }_x$. For a nonzero one-variable polynomial $P$ and $v\in {\mathcal{E}}^{\prime }(\mathbb{R})$, a compactly supported solution to $P(D)u=v$ exists exactly when $v$ annihilates every smooth solution of the transposed equation $P(-D)\phi =0$. For simple roots ${\alpha }_j$, these test solutions span the exponentials $e^{-i{\alpha }_{j}x}$, so the condition is $\hat{v}({\alpha }_j)=0$. Dividing the entire transform by $P$ and applying polynomial division preservation of exponential type and the Paley–Wiener–Schwartz theorem constructs the compactly supported solution. It is unique because an entire function killed by a nonzero polynomial must vanish identically. For a root of multiplicity $r$, the conditions become ${\hat{v}}^{(k)}(\alpha )=0$ for $0\le k<r$, corresponding to $x^k{e}^{-i\alpha x}$.