 Oscillatory-root criterion for a real second-order equation

For real constants with $a_2\ne 0$, a nonzero solution of the homogeneous linear differential equation $a_2{f}^{\prime \prime }+a_1{f}^{\prime }+a_{0}f=0$ has a sinusoidal factor exactly when the discriminant $a_1^2-4a_0{a}_2$ is negative. Indeed the characteristic polynomial has roots $-a_1/(2a_2)\pm i\omega$, where $\omega =\sqrt{4a_0{a}_2-a_1^2}/(2|a_2|)$, giving $f=e^{-a_{1}t/(2a_2)}(C_1\cos \omega t+C_2\sin \omega t)$. Real or repeated roots instead give real exponential functions, possibly multiplied by a linear polynomial. Oscillatory behaviour and decay are separate: decay requires $a_1/a_2>0$.