Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 2 / 12 / ii / Solution

The damped angular frequency is $\Omega =\sqrt{k-{\lambda }^2/4}$. The oscillation envelope is proportional to $e^{-\lambda t/2}$. A decrease by a quarter leaves three quarters of the original amplitude, so $e^{-\lambda t/2}=3/4$. ThereforeThis concerns the envelope; it does not ask when the instantaneous displacement first equals three quarters of a chosen value.