I'm struggling with understending of how should I solve differential equations in distributions. For example,if I'm asked to solve in $\mathcal D'$ the following equation: $$ F'-2F=0$$ Okay, so $F'=2F$ in terms of distributions, which means $F'$ and $2F$ act the same on every test function. Recalling the definition: $$\langle F', \varphi \rangle = - \langle F, \varphi' \rangle \ \ \ \ \ \ \ \langle 2F, \varphi \rangle = \langle F, 2\varphi \rangle$$ So should I write it in the way $$ -\int\limits_{\mathbf R} F\varphi'\ dx =2\int\limits_{\mathbf R} F\varphi\ dx $$ But it feels like I go back from definition and losing solutions which do not have this form.
Reading about distributions I found out that sometimes problems are solved with "well-known" little facts, perhaps I miss something like that. I'm sure I have to reduce this equation to something known, like, $F'=\delta(x)$, but I don't see how I can do that.
So basically I'm looking for hints in techniques or facts/tricks that could help me to solve this and other linear DE in distributions. Thanks!
Multiply the equation with $e^{-2x}$. Then you can rewrite the left hand side as the derivative $(e^{-2x}F)'$ and shall solve $u'=0$ where $u=e^{-2x}F.$