If $x \log x \frac{dy}{dx} + y = \log x^2$ and $y(e) = 0$,then $y(e^2) = $

Let $y=Y(x)$ be the solution of the differential equation $\frac{dy}{dx}+y \tan x=2x+x^2 \tan x$,$x \in \left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$,such that $Y(0)=1$,then

Let $y=y(x)$ be the solution of the differential equation $(x+1) y^{\prime}-y=e^{3 x}(x+1)^{2}$,with $y(0)=\frac{1}{3}$. Then,the point $x=-\frac{4}{3}$ for the curve $y = y ( x )$ is

The integrating factor of the differential equation $\frac{dy}{dx} = y \tan x - y^2 \sec x$ is

If $f: R \rightarrow R$ is a differentiable function such that $f^{\prime}(x) > 2f(x)$ for all $x \in R$,and $f(0) = 1$,then:

If $y = y(x)$ is the solution of the differential equation $\sqrt{4-x^2} \frac{dy}{dx} = \left(\left(\sin^{-1}\left(\frac{x}{2}\right)\right)^2 - y\right) \sin^{-1}\left(\frac{x}{2}\right)$ for $-2 \leq x \leq 2$ with $y(2) = \frac{\pi^2-8}{4}$,then $y^2(0)$ is equal to