 Distributional regularity of a constant-coefficient ordinary differential equation

For a nonzero constant-coefficient polynomial differential operator in one variable, every homogeneous distribution solution is a classical solution, indeed analytic. Factor the polynomial into powers of distinct characteristic roots of a constant-coefficient differential equation. The kernel decomposition for coprime polynomials reduces the equation to $(D-\lambda {)}^{m}v=0$, and multiplication of a distribution by a smooth function reduces this to $D^m(e^{-\lambda x}v)=0$. The fact that a distribution with zero derivative is constant, iterated, makes the latter distribution a polynomial of degree less than $m$.