Is there a unique solution of the first-order linear differential equation with constant factor apart from the uncertainty of the integration factor?

Question

Consider the following differential equation $\dot{y}=ay$, where $y:\mathbb{R}\to \mathbb{R}$ and $a\in \mathbb{R}$. I would like to know if the function $c\exp \left( ax \right)$ is the unique solution of this equation, where $a\in \mathbb{R}$ is the integration factor. If yes, how to prove it?

Answer 1

Let $f$ be a solution of the differental equation and define

$y(x):=\frac{f(x)}{e^{ax}}$. Show that $f'(x)=0$ for all $x$. Thus: $f$ is constant