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For a genuinely second-order linear ordinary differential equation, assume $a_2\ne 0$. Substituting the exponential function $f=e^{rt}$ into the homogeneous linear differential equation gives $a_2{r}^2+a_{1}r+a_0=0$. Oscillations arise precisely when its two roots of a polynomial have nonzero imaginary parts, namelyIn that case, writing $\omega =\sqrt{4a_0{a}_2-a_1^2}/(2|a_2|)>0$, the real complementary function isEvery 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 $a_1\ne 0$, only real exponential functions occur. If $a_2=a_1=0$ and $a_0\ne 0$, 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.