Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 2 / 5C (Differential Equations)

$y_0\prime \prime =(x^2-1)y_0$, so $\alpha =1$.

Substitution and cancellation give $f\prime \prime -2xf\prime +(\alpha -1)f=0$.

Equating powers gives $(n+2)(n+1)a_{n+2}+(\alpha -1-2n)a_n=0$.

$y\prime (0)=0$ forces the odd series to vanish. The even recurrence terminates at degree $2N$ precisely when $\alpha =4N+1$.

After division by $e^{-x^2/2}$, $f\prime \prime -2xf\prime +8f=2$. The initial data and evenness give $f=1-3x^2+x^4$, hence $y=e^{-x^2/2}(1-3x^2+x^4)$.