Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 8 / ii / Solution

Summing the geometric series givesThere is a particularly short verification using the product rule:Also $y(0)=1$ and $y^{\prime }(0)=0$. The radius of convergence of the series is one, set by the three complex cube roots of unity; the closed expression continues the solution beyond that disk wherever its denominator is nonzero. The real initial-value solution extends on the interval $(-\infty ,1)$.