Let \( g \) be a differentiable function such that \( \int_{0}^{x} g(t)\,dt=x-\int_{0}^{x} t g(t)\,dt,\, x\ge 0 \) and let \( y=y(x) \) satisfy the differential equation \( \dfrac{dy}{dx}-y\tan x=2(x+1)\sec x\, g(x),\, x\in\left[0,\dfrac{\pi}{2}\right] \). If \( y(0)=0 \), then \( y\!\left(\dfrac{\pi}{3}\right) \) is equal to:
