Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2020 / Paper 1 / 4 / b / Solution

For a Frobenius exponent $r$, put $p_0={\lim }_{x\to 0}xP(x)=1$ and $q_0={\lim }_{x\to 0}{x}^{2}Q(x)=0$. The indicial equation issoA fundamental pair has the repeated-root Frobenius formon a positive-$x$ interval. In this case a real analytic solution is particularly simple: $y_1=1-x$ satisfies the equation exactly. A convenient logarithmic companion, obtained below, is $y_2=x+(1-x)\ln [x/(1-x)]$ for $0<x<1$. It is independent of $y_1$ because of its logarithmic divergence at zero.