I want to solve distributional equations of the form $fT=\delta_0$ for a $C^\infty$ function $f$.
For the equation $fT=0$, we can bound the support of $T$ by $$\text{supp }T\subset f^{-1}(\{0\})$$ and that usually helps solving such equations.
However, for $fT=\delta_0$ I don't seem to understand a pattern. For example, let's understand the solution of $xT=\delta_0$:
We want to find a distribution $T$ such that $T(x\varphi)=\varphi(0)$ for every $\varphi\in\mathcal{D}(\mathbb{R})$. Since we can't evaluate $\varphi(x)/x$ at $0$, the next logical thing to do is to define $T$ as $$T(\varphi)=\lim_{x\to 0}\frac{\varphi(x)}{x}.$$ Since $\varphi$ is $C^\infty$, this works if and only if $\varphi(0)=0$. We can fix this by defining $$T(\varphi)=\lim_{x\to 0}\frac{\varphi(x)-\varphi(0)}{x}=\varphi'(0).$$ That is, $-\delta'_0$ is a particular solution of the equation $xT=\delta_0$.
Now, notice that this method cant be generalized to solve equations of the form $fT=\delta_0$ if $f$ is equal to $e^{-1/x}$ for $x\geq 0$ and $0$ otherwise because $f^{(k)}(0)=0$ for all $k$.
What should we do in this case? Is there a good method for solving general equations of the form $fT=\delta_0$?