Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 2 / 5C (Differential Equations)

For a normalized equation$x_0$ is an ordinary point when $P,Q$ are analytic there. It is a regular singular point when $(x-x_0)P(x)$ and $(x-x_0{)}^{2}Q(x)$ are analytic there. These are the ordinary point criterion for a second-order equation and regular singular point criterion for a second-order equation.

Hereso $x=0$ is regular singular. Seek a power-series solution of a differential equationEquating the coefficient of $x^n$ givesand henceThereforeThe series terminates exactly when $\lambda =N$ is a nonnegative integer: the factor with $j=N$ then makes $a_{N+1}=0$. Thus the polynomial solutions occur for

After division by $x$, the coefficient of $y^{\prime }$ is $P(x)=1/x-1$. The Abel identity for the Wronskian givesso

For $\lambda =1$, direct substitution confirms that $y_1=1-x$. Reduction of order gives a second solution proportional toTo extract the requested coefficients, putFor the differential operatorone findsThe analytic correction $h={\sum }_{n\ge 1}{b}_n{x}^n$ must therefore satisfy $L[h]=3-x$. Its constant and linear coefficients giveHence