Geometric effects of parameters in homogeneous second-order ordinary differential equations with constant coefficients

Question

In homogeneous second-order ordinary differential equations (ODE) with constant coefficients, of the form: \begin{equation} ay''+by'+cy=0, \end{equation} is there any conclusion about the effects of each of the parameters $a$, $b$ and $c$ on the graph of an ODE solution? It's a curiosity I've been having, but I've never seen anything like this in ODE books.

Answer 1

Let the independent variable in your equation be the time $t$. The equation can be written in the form

$$ y''+\gamma y'+\omega_0^2 y=0,\quad \gamma=\frac{b}{a},\quad \omega_0^2=\frac{c}{a}.\tag{*} $$

I assume $\gamma,\omega_0^2>0$, since this is the case of interest in physical problems. Then $(*)$ is Newton's second law for a damped harmonic oscillator. There is a competition between the two frequencies $\gamma$ and $\omega_0$: larger values of $\omega_0$ promote longer-lived oscillations in the solutions to $(*)$, while larger values of $\gamma$ "dampen" the oscillations; i.e. cause them to decay towards zero.