Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 8 / b / iv / Solution

Write $t=1-x$. The analytic solution has $y_1=a_0+O(t)$ with $a_0\ne 0$, so the integrand in the reduction of order formula satisfiesIntegrating with respect to $x$ gives $-a_0^{-2}\log (1-x)+O(1)$. Multiplication by $y_1$ therefore givesAny solution independent of $y_1$ has a nonzero coefficient of this second solution. HenceAn added multiple of $y_1$ stays bounded and cannot cancel this logarithm. On the real side $x>1$, the corresponding statement uses $\log |1-x|$.