Let g be a differentiable function such that ∫_0^x g(t) d t=x-∫_0^x t g(t) d t, x ≥ 0 and let y=y(x) satisfy the differential equation (d y)/(d x)-y…

Let $\displaystyle g$ be a differentiable function such that $\displaystyle \int_0^x g(t) d t=x-\int_0^x t g(t) d t, x \geq 0$ and let $\displaystyle y=y(x)$ satisfy the differential equation $\displaystyle \frac{d y}{d x}-y \tan x=2(x+1) \sec x g(x), x \in\left[0, \frac{\pi}{2}\right)$. If $\displaystyle y(0)=0$, then $\displaystyle y\left(\frac{\pi}{3}\right)$ is equal to