How to solve a simple differential equation in the way of weak solutions?

Question

I want to prove that the solution equation

$y'=y$

is $y=Ce^x$, where $C$ is a constant. Here $y$ belongs to the space of linear operators on $C_0^\infty(\mathbb{R})$, and $y'$ is its weak derivative.

Then using the definition of weak derivative, we get for any test function $\varphi$ whose support is compact on $\mathbb{R}$, the following equation holds

$\int_{-\infty}^{\infty}y(x)\varphi(x)dx=\int_{-\infty}^{\infty}y'(x)\varphi(x)dx=-\int_{-\infty}^{\infty}y(x)\varphi'(x)dx$

Then we need to solve $\int_{-\infty}^{\infty}y(x)\varphi(x)dx=-\int_{-\infty}^{\infty}y(x)\varphi'(x)dx$, but I am stuck on how to solve such type of equation, any advice would be appreciated!

Answer 1

I don't think that's the good way to go. It is better to mimic the computation that one usually does with the standard concept of solution, that is, multiply termwise by the integrating factor $e^{-x}$.

The distributional calculus has the same rules as the ordinary calculus, and in particular one has both Leibniz's rule and the fact that $$ T'=0\quad \iff\quad T\ \text{is the constant distribution}, $$ so everything is going to work out correctly.