We have the following result about differential equation of order $2$ with analytical coefficients:
Consider the equation $y''+ay'+by=0$, where $a$ and $b$ are analytical in a neighborhood of $t_0$, with respective radius of convergence $R_1$ and $R_2$. Then any solution of this equations is analytical in a neighborhood of $t_0$, with radius of convergence $\geq \min(R_1,R_2)$.
(to prove this, we use the definition of radius of convergence of $\sum_n a_nx^n$: it's the supremum of the $M$ such that $\{a_nM^n\}$ is bounded)
Since $\frac 1{\sqrt t}=\frac 1{\sqrt{t_0+t-t_0}}=\frac 1{\sqrt{t_0}}\frac 1{\sqrt{1+\frac{t-t_0}{t_0}}}$, and since the power series of $\frac 1{\sqrt{1+u}}$ has a radius of convergence $1$, $\frac 1{\sqrt{t_0}}$ has a radius of convergence of $t_0$. So the solution of $y''(t)+\frac 1{\sqrt t}y(t)=0$ which satisfies $y(t_0)=0$ and $y'(t_0)\neq 0$ has a radius of convergence $\geq t_0$.