I am having a bit of trouble solving distributional equations. An example that I am currently working on is to show that the distributional equation $T.x = 1$ has a solution if and only if $T=p.v.(\frac{1}{x})+c\delta_{0}$.
A further piece of information that I am given is that every distribution $T \in D'(\mathbb{R}^{n})$ with support $supp(T) \subset {0}$ has the form: $$T=\sum_{|\alpha|\leq N}c_{\alpha}(\partial^{\alpha}\delta_{0})$$ with $c_\alpha \in \mathbb{R}$ for some $N \in \mathbb{N}$.
I have shown that the proposed $T$ is indeed a solution. It remains to show the other implication, where I suspect the hint should help. So $T.x = 1$ means that $<T,x\phi>=\int_{\mathbb{R}} \phi dx$ for all test functions $\phi$. Since we need to manipulate the integral to get an integrand involving an $x\phi$ term, I guess we could split the integral so that $$<T,x\phi>=\int_{B(0,\epsilon)}\phi dx + \int_{\mathbb{R} \setminus B(0,\epsilon)}\frac{x\phi}{x}dx = \int_{B(0,\epsilon)}\phi dx+ <p.v.(\frac{1}{x}),x\phi>$$
So now we have the principal value term, but I'm not sure how to proceed to get the $\delta_{0}$ component! Would love some help!