Coordinate-degenerate constant-coefficient differential equation ($xP(D)u=0$)

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 $P(D)(xu)$.