I'm trying to understand differentiability and to do so i'm trying to answer a question but cannot work it out:
(a). Suppose that $f$ : $\mathbb{R}$ $\rightarrow (0, \infty)$ is differntiable and satisfies f'(x) = af(x) for all $x \in \mathbb{R}$, for some constant $a \in \mathbb{R}$. Prove that $f(x) = Ce^{ax}$, for some constant $C \in \mathbb{R}$.
(b). Now suppose that $h:\mathbb{R} \rightarrow \mathbb{R}$ is twice differentiable and, for all $x \in \mathbb{R}$
h''(x) + 3h'(x) + 2h(x) = 0.
Let F(x) = h'(x) + 2h(x) and G(x) = h'(x) + h(x). Show that $F$ and $G$ both satisfy the hypotheses of part (a). Hence prove that:
$h(x) = Ce^{-x} + De^{-2x}$
Any help would be appreciated.