Mathematics Question from JEE Main 2024 · 30 January, Shift 1

Let $y = y(x)$ be the solution of the differential equation $(1 - x^2)\,dy = \left[xy + (x^3 + 2)\sqrt{3(1 - x^2)}\right] dx$, $-1 < x < 1$, $y(0) = 0$. If $y\left(\frac{1}{2}\right) = \frac{m}{n}$, where $m$ and $n$ are co-prime numbers, then $m + n$ is equal to _____.