The solution of the differential equation $x \log x \frac{dy}{dx} + y = 2 \log x$ is

Let $y(x)$ be the solution of the differential equation $(x \log x) \frac{dy}{dx} + y = 2x \log x$,$(x \ge 1)$. Then $y(e)$ is equal to: $[y(1) = 0]$

The general solution of the differential equation $(y^2 - x^3) dx - xy dy = 0, (x \neq 0)$ is (where $c$ is a constant of integration)

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

Let $y=y(x)$ be the solution curve of the differential equation $\frac{dy}{dx}+\frac{1}{x^{2}-1}y=\left(\frac{x-1}{x+1}\right)^{\frac{1}{2}}$,$x>1$ passing through the point $\left(2, \sqrt{\frac{1}{3}}\right)$. Then $\sqrt{7}y(8)$ is equal to.

Let $y=y(x)$ be the solution of the differential equation,$x y^{\prime}-y=x^{2}(x \cos x+\sin x), x>0$. If $y(\pi)=\pi$,then $y^{\prime \prime}\left(\frac{\pi}{2}\right)+y\left(\frac{\pi}{2}\right)$ is equal to