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The characteristic equation $ar^2 + br + c = 0$ is solved by applying the quadratic formula. This yields the roots $r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, which provide the values for the parameter $r$ in the exponential trial solution.

Conditions: The characteristic equation is quadratic in $r$.; $a \neq 0$.

The characteristic equation $ar^2 + br + c = 0$ is solved by applying the quadratic formula. This yields the roots $r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, which provide the values for the parameter $r$ in the exponential trial solution.

Conditions: The characteristic equation is quadratic in $r$.; $a \neq 0$.