Existence, Uniqueness & Bifurcations · Superchallenge problem

For a constant $c$, consider $y' = y^2 + \dfrac{c}{t^2}$ on $t \ge 1$. There is a critical value $c^*$ such that some solution exists on all of $[1, \infty)$ when $c \le c^*$, while every solution blows up at a finite time when $c > c^*$. Find $c^*$.