Differential Equations

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