If $\tan x$ is an integrating factor of the differential equation $\frac{dy}{dx} + Py = Q$, then $P$ can be

If the solution curve of the differential equation $(y-2 \ln x) dx + (x \ln x^2) dy = 0, x > 1$ passes through the points $(e, \frac{4}{3})$ and $(e^4, \alpha)$,then $\alpha$ is equal to $................$.

If $y = y(x)$ is the solution of the differential equation $\frac{dy}{dx} + (\tan x)y = \sin x, 0 \leq x \leq \frac{\pi}{3},$ with $y(0) = 0,$ then $y\left(\frac{\pi}{4}\right)$ is equal to:

If $y=y(x)$ is the solution curve of the differential equation $\frac{dy}{dx} + y \tan x = x \sec x$,$0 \leq x \leq \frac{\pi}{3}$,with $y(0)=1$,then $y\left(\frac{\pi}{6}\right)$ is equal to

The general solution of the differential equation $(\sec x + \tan x) \frac{dy}{dx} + (\sec^2 x + \sec x \tan x) y = 1$ 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 $.........$.