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 / iii / Solution

For the normal-form coefficient $p(x)=-1/(1-x)$, Abel's identity gives the WronskianUse reduction of order: if $y_2=y_{1}v$, then $W=y_1^2{v}^{\prime }$. Absorbing a nonzero Wronskian constant into the second integration constant, the general solution isThis local representation applies on intervals where $y_1\ne 0$, in particular sufficiently near $x=1$ because $y_1(1)=a_0\ne 0$.