Let y=y(x) be the solution of the differential equation (1-x^2) d y=(x y+(x^3+2) √(1-x^2)) d x,-1<x<1, and y(0)=0. If ∫_(-1/2)^(1/2) √(1-x^2) y(x) d…

Let $\displaystyle y=y(x)$ be the solution of the differential equation $$\left(1-x^2\right) d y=\left(x y+\left(x^3+2\right) \sqrt{1-x^2}\right) d x,-1<x<1 \text {, and } y(0)=0 \text {. If } $$ $\displaystyle \int_{\frac{-1}{2}}^{\frac{1}{2}} \sqrt{1-x^2} y(x) d x=k$, then $\displaystyle k^{-1}$ is equal to \_\_\_\_.