Definition:Characteristic Equation of ODE

Definition
Let:


 * $a y + b y' + c y = f\left({x,y,y',y}\right)$

be a second order linear ordinary differential equation, where $a,b,c$ are constant.

The characteristic equation of this ODE is:


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

where $r$ satisfies $y = e^{rx}$.

Here $e^x$ is the real exponential.