differential equations - Elegant way of obtaining the envelope of oscillating function

I am solving a differential equation numerically and the output is an oscillating function with the amplitude of the oscillation decaying in time. I would like to extract the power law governing this amplitude.

Let's make this concrete. Let's say I am solving the system

x''[t] + 2t^(-1) x'[t] + t^(-2) x[t] == 0

which has solutions

x= C[1] t^(-1/2) Cos[Sqrt[3]/2 Log[t]]

and the corresponding $\sin$. I would like to extract the power law $t^{-1/2}$.

Of course, my system is much more complicated and I have to solve it numerically. I get as my solution an interpolating function. I was thinking of generating a table of function values, sorting according to the maxima and then fitting a power law, but maybe there is a better way?

Thanks!