The integrating factor of $\left(x+2 y^3\right) \frac{d y}{d x}=y^2$ is

The general solution of $\frac{dy}{dx} + y \tan x = 2x + x^2 \tan x$ is:

The solution of $\cos y + (x \sin y - 1) \frac{dy}{dx} = 0$ is

The solution of $\frac{dy}{dx} + \frac{1}{3}y = 1$ is

Let $y=y(x)$ be a solution of the differential equation $(x \cos x) dy + (xy \sin x + y \cos x - 1) dx = 0$,$0 < x < \frac{\pi}{2}$. If $\frac{\pi}{3} y(\frac{\pi}{3}) = \sqrt{3}$,then $|\frac{\pi}{6} y''(\frac{\pi}{6}) + 2 y'(\frac{\pi}{6})|$ is equal to $.........$.

Let $y = y(x)$ be the solution of the differential equation $x\sqrt{1-x^2} dy + (y\sqrt{1-x^2} - x\cos^{-1}x) dx = 0$, where $x \in (0, 1)$ and $\lim_{x\to 1^-} y(x) = 1$. Then $y\left(\frac{1}{2}\right)$ equals: