The solution of the differential equation $2x \frac{dy}{dx} - y = 3$ represents a family of

If $x \, dy = y(dx + y \, dy)$,$y(1) = 1$,$y(x) > 0$,then $y(-3)$ is

The integrating factor of the differential equation $x \frac{dy}{dx} + y \log x = x e^x \cdot x^{-1/2} \log x$ for $x > 0$ is:

Let $y=y(x), x>1$,be the solution of the differential equation $(x-1) \frac{d y}{d x}+2 x y=\frac{1}{x-1}$,with $y(2)=\frac{1+e^{4}}{2 e^{4}}$. If $y(3)=\frac{e^{\alpha}+1}{\beta e^{\alpha}}$,then the value of $\alpha+\beta$ is equal to

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

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