Alibaba Global Mathematics Competition

Firstly, $x^{j} f_{k}(x)(0 \leq j \leq 2 k)$ are all absolutely integrable when $k \geq 1$. Then, by (6), $y=\left(S f_{k}\right)(x)$ is a $2 k$-th order continuous differentiable function. By Lemma 0.1 and Lemma $0.4, x y^{\prime \prime}+c_{1} y^{\prime}+c_{2} x y=0$ is equivalent to $\left(x^{2} f_{k}^{\prime}+2 x f_{k}\right)-c_{1} x f_{k}-\frac{c_{2}}{4 \pi^{2}} f_{k}^{\prime}=0$. Inputting $f_{k}(x)=\left(1+x^{2}\right)^{-1-k}$, we get $c_{1}=-2 k$ and $c_{2}=-4 \pi^{2}$.