Past exam of the mathematical methods I course of the University of Oxford / 2018 / 8 / b / Solution

Substitute the power series and its termwise derivatives into the ordinary differential equation. The coefficient of $x^n$, for $n\ge 0$, isThe bracket factors as $(n-3)(n-4)$, so the power-series solution of a differential equation satisfiesThe odd and even recurrence relations terminate: $a_3=-a_1$, $a_5=0$, and $a_2=-6a_0$, $a_4=a_0$, $a_6=0$. Thus the full solution is $y=a_0(1-6x^2+x^4)+a_1(x-x^3)$. The two polynomials have independent initial data at zero, and the leading coefficient $1+x^2$ never vanishes on the real axis, so every real solution of a differential equation for this equation is their linear combination on the entire real axis. The given initial data select $a_0=0$, $a_1=1$: