Suppose $y(t)$ is a solution of $\frac{d^2y}{dt^2} +\frac{ay}{t^2} =0$ on$(0,\infty)$
If $0<a<\frac{1}{4}$ ,then $\lim_{t\to +0} y(t)=0$
I tried solve this equation, but it is difficult. So I tried to find $f(t)$ s.t., $y(t)f(t) \to 0$ and $f(t) \not \to 0$, but I couldn't find it.
Let $c_i $is solutions of $c^2-c+a=0$, $y_i=x^{c_i}$ So, we determine $y(t_0)$ and $y'(t_0)$, by uniqueness of Cauchy problem, $\{$solution$\}=$span$\{y_1, y_2\}$