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

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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 $DE = \delta$ if and only if $E = E_* + T_F$ where:

$DE_* = \delta$
$DF = 0$


Proof

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:

$E = E_* + T_F$

$\Box$


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:

\(\ds D E\) \(=\) \(\ds D E_* + D T_F\)
\(\ds \) \(=\) \(\ds \delta + \mathbf 0\)
\(\ds \) \(=\) \(\ds \delta\)

$\blacksquare$


Sources

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