= Solution
For a genuinely second-order <linear ordinary differential equation>, assume $a_2\ne0$. Substituting the <exponential function> $f=e^{rt}$ into the <homogeneous linear differential equation> gives $a_2r^2+a_1r+a_0=0$. Oscillations arise precisely when its two <roots of a polynomial> have nonzero <imaginary parts>, namely
$$
\boxed{a_1^2-4a_0a_2<0.}
$$
In that case, writing $\omega=\sqrt{4a_0a_2-a_1^2}/(2|a_2|)>0$, the real <complementary function> is
$$
f(t)=e^{-a_1t/(2a_2)}\bigl(C_1\cos\omega t+C_2\sin\omega t\bigr).
$$
Every nonzero such solution has an oscillatory factor; the oscillation may decay, grow, or keep constant amplitude according to the sign of $a_1/a_2$. This is the <oscillatory-root criterion for a real second-order equation>; decay is not a prerequisite for oscillation.
For completeness, if $a_2=0$ and $a_1\ne0$, only real <exponential functions> occur. If $a_2=a_1=0$ and $a_0\ne0$, only $f=0$ is allowed. If all three coefficients vanish, every twice-differentiable function is allowed, including oscillatory ones; the second-order criterion does not apply to this degenerate identity.
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