= Oscillatory-root criterion for a real second-order equation
{title2=$a_1^2-4a_0a_2<0$}
For real constants with $a_2\ne0$, a nonzero solution of the <homogeneous linear differential equation> $a_2f''+a_1f'+a_0f=0$ has a sinusoidal factor exactly when the <discriminant> $a_1^2-4a_0a_2$ is negative. Indeed the <characteristic polynomial> has roots $-a_1/(2a_2)\pm i\omega$, where $\omega=\sqrt{4a_0a_2-a_1^2}/(2|a_2|)$, giving $f=e^{-a_1t/(2a_2)}(C_1\cos\omega t+C_2\sin\omega t)$. Real or repeated roots instead give real <exponential functions>, possibly multiplied by a linear <polynomial>. Oscillatory behaviour and decay are separate: decay requires $a_1/a_2>0$.
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