Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 3 / b / ii / Solution

The PDF bounds the absolute value of the forcing. Taking absolute values in the Green function integral and using $|\sin |\le 1$ yieldsFor $a\ne 1$, this givesThe ratio is nonnegative for either $a<1$ or $a>1$. The missing special case is $a=1$, for whichThis proves decay of a damped oscillator under exponentially decaying forcing. A one-sided upper bound on $g$ without an absolute value, as in the transcription, would not justify the conclusion.