First order distribution differential equation

Question

I am struggling to solve this differential equation where T is a distribution.

$$ T' -T = \delta_0$$

I know a method to solve differential equations of the form $ T' +bT = T_f$ where $T_f$ is a regular distribution, but since $ \delta_0$ is not a regular distribution, I don't think I can apply that theorem.

I also can't apply the theorem I would for differential equations of the form $aT'' + bT' +cT = \delta_{x_0}$ Since it would need $a \neq 0$.

I feel like it's easy and I'm not seeing the obvious answer.

Answer 1

Pretend that $T$ and $\delta_0$ are regular functions. Then, using standard methods, you can show that the solution to the differential equation $T'-T=\delta_0$ is $$ T(x)=T(x_0)e^{x-x_0}+e^x\int_{x_0}^x e^{-x'}\delta_0(x')\,dx'. $$ This solution is also valid if $T$ and $\delta_0$ are distributions.

Answer 2

Multiply the equation with the integrating factor $e^{-x}$ so that the left hand side becomes $(e^{-x}T)'$ and the right hand side becomes $e^{-x}\delta_0=\delta_0.$