I'm learning the basics of distributional theory. Reading a book, I've found this exercises: Find the general solution of
$$(x-1)T= \delta$$
$$(x-1)T =\delta'$$
I've tried different starting points but no solutions. Please, can you help me?
2026-03-30 05:23:30.1774848210
Distributional differential equations (delta)
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Hints: So the thing to know is that solutions to $$ x\,T = f $$ are of the form $g + C\,\delta_0$ where $g$ is any particular solution of the equation. Replacing $x$ by $x-1$, this is still true but with $\delta_1$.
To find a particular solution, think about the fact that the Dirac is only non-zero at $x=0$ ... so look at what happens if your replace $x$ by $0$.