The solution of the differential equation $2 x \left(\frac{d y}{d x}\right) - y = 4$ represents a family of

If $y=y(x)$ satisfies $\left(\frac{2+\sin x}{1+y}\right) \frac{dy}{dx} = -\cos x$ such that $y(0)=2$,then $y\left(\frac{\pi}{2}\right)$ is equal to

The particular solution of the differential equation $y(1+\log x) \frac{dx}{dy} - x \log x = 0$ when $x=e, y=e^2$ is

The general solution of the differential equation $(x-y)^2 \frac{dy}{dx} = a^2$ is

The solution of the differential equation $\frac{dx}{dy} + 2yx = 2y$ which passes through the point $(2,0)$ is

Let $y=y(x)$ be the solution of the differential equation $(1+y^2) e^{\tan x} dx + \cos^2 x(1+e^{2 \tan x}) dy = 0$,with $y(0)=1$. Then $y(\frac{\pi}{4})$ is equal to: