The solution of the differential equation $\frac{dy}{dx} - y \tan x = e^x \sec x$ is

If $y=y(x)$ is the solution of the differential equation $\frac{dy}{dx} + 2y \tan x = \sin x$ with the condition $y(\frac{\pi}{3}) = 0$,then the maximum value of the function $y(x)$ over $\mathbb{R}$ is equal to:

Let $y=y(x)$ be the solution of the differential equation $\cos x(\ln(\cos x))^2 dy + (\sin x - 3y \sin x \ln(\cos x)) dx = 0$,where $x \in (0, \frac{\pi}{2})$. If $y(\frac{\pi}{4}) = \frac{-1}{\ln 2}$,then $y(\frac{\pi}{6})$ is:

If the solution curve of the differential equation $(\tan^{-1} y - x) dy = (1 + y^2) dx$ passes through the point $(1, 0)$,then the abscissa of the point on the curve whose ordinate is $\tan(1)$ is

The solution of the equation $\frac{dy}{dx} + 2y \tan x = \sin x$ satisfying $y = 0$ when $x = \frac{\pi}{3}$ is:

If $\cos x \frac{dy}{dx} = y \sin x - 1$, where $x \neq (2n+1) \frac{\pi}{2}, n \in Z$, is the differential equation corresponding to the curve $y = f(x)$ and $f(0) = 1$, then $f(x) =$