We focus on the distribution of the extinction times of a population whose size N(t) follows a Feller diffusion process with inhomogeneous growth term r (t). Obtaining a precise description of the extinction times and of their dependence with respect to r (t) has important applications in adaptive biology, for understanding "evolutionary rescue" phenomena. A formula for the distribution of the extinction times has been recently obtained, through probabilistic arguments. The aim of this note is to propose a new derivation of this formula, based on the analysis of a degenerate parabolic partial differential equation.