Consider the homogeneous equation

$$ay''+by'+cy=0,$$

with constant coefficients and $a\ne0$.

Looking for exponential solutions $y=e^{rt}$ gives the characteristic equation

$$ar^2+br+c=0.$$

Its roots determine the solution form.

For distinct real roots $r_1,r_2$,

$$y=C_1e^{r_1t}+C_2e^{r_2t}.$$

For a repeated root $r$,

$$y=(C_1+C_2t)e^{rt}.$$

For complex-conjugate roots $r=\alpha\pm i\beta$,

$$y=e^{\alpha t}\bigl(C_1\cos(\beta t)+C_2\sin(\beta t)\bigr).$$

The characteristic polynomial converts a differential equation into an algebraic root problem; the root geometry then determines growth, decay and oscillation.