Fundamental Solutions to Distributional Homogeneous ODE with Constant Coefficients differ by Classical Solution

Theorem
Let $E_*, E, T \in \map {\DD'} \R$ be distributions.

Let $D$ be an ordinary differential operator with constant coefficients.

Let $f$ be a function differentiable by $D$.

Let $T_f \in \map {\DD'} \R$ be a distribution associated with $f$.

Let $\delta$ be the Dirac delta distribution.

Let $E_*$ be the fundamental solution to $D E_* = \delta$

Then $E$ is a fundamental solution to $DE = \delta$ $E = E_* + T_F$ where:


 * $DE_* = \delta$


 * $DF = 0$

Necessary Condition
Suppose both $E_*$ and $E$ are fundamental solutions:


 * $DE = \delta$


 * $DE_* = \delta$

Taking the difference yields:


 * $D \paren {E - E_*} = \mathbf 0$

where $\mathbf 0 \in \map {\DD'} \R$ is the zero distribution.

By Solution to Distributional Ordinary Differential Equation with Constant Coefficients:


 * $E - E_* = T_F$

where $DF = 0$.

Hence:


 * $E = E_* + T_F$

Sufficient condition
Suppose $F$ is a classical solution to $DF = 0$.

That is, suppose:


 * $DT_F = \mathbf 0$

Suppose $E_* \in \map {\DD'} \R$ is a fundamental solution to $DE_* = \delta$.

Let $E$ be a distribution such that $E:= E_* + T_F$.

Then: