= Coordinate-degenerate constant-coefficient differential equation
{title2=$xP(D)u=0$}
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 $P(D)(xu)$.
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