Distributional regularity of a constant-coefficient ordinary differential equation ($P(D)v=0⟹v\in C^{\omega }$)

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 $m$.

For a positive-order constant-coefficient ordinary differential equation, the kernel of multiplication by a coordinate gives $xP(D)u=0$ exactly when $P(D)u=C{\delta }_0$. Hence every solution is $u=v+CE$, where $P(D)v=0$ and $E$ is any fundamental solution of a linear differential operator. Taking a retarded fundamental solution of a constant-coefficient ordinary differential operator shows that the extra freedom is a single jump in the derivative of order one below the operator order. The coordinate multiplies the already differentiated distribution: this is a different operator from $P(D)(xu)$.

If a degree-$n$ characteristic polynomial has distinct roots of a polynomial ${\lambda }_j$ of multiplicities $m_j$, all homogeneous distribution solutions are the displayed combinations. There are ${\sum }_j{m}_j=n$ independent coefficients. Repeated characteristic roots of a constant-coefficient differential equation account for the polynomial factors multiplying exponentials; conjugate pairs give real sine and cosine combinations when real solutions are desired.