In my notes, I found that the following equation
$$\frac{d^2}{d x ^2} \varphi(x) + h \frac{d}{d x} \varphi(x) + k \varphi(x) = 0$$
with $x \in \mathbb{R}$, has the general solution:
$$\varphi(x) = A_1 e^{r_1 x} + A_2 e^{r_2 x}$$
where $r_{1,2}$ are the roots of the characteristic equation
$$r^2 + hr + k = 0$$
1) Is it correct?
2) How is the characteristic equation related to the original equation?
Google results give many different forms of the above equations, with different notations, and the following case is never clearly showed: let's consider $h = 0$, so the Helmholtz equation case, used in Physics as a wave equation:
3) If $k < 0$ and so $r_{1,2}$ are real, what about $C_1$ and $C_2$? And how can their values be determined?
Have a look at
https://en.wikipedia.org/wiki/Harmonic_oscillator
specifically to the 'Universal Oscillator Equation' section