If $c$ is any arbitrary constant,then the general solution of the differential equation $ydx - xdy = xy\,dx$ is given by

If $2x - y + c \log(x - 2y - 4) = k$ is the general solution of $\frac{dy}{dx} = \frac{2x - 4y - 5}{x - 2y + 2}$,then $c =$

If the solution $y(x)$ of the given differential equation $(e^y+1) \cos x \, dx + e^y \sin x \, dy = 0$ passes through the point $(\frac{\pi}{2}, 0)$,then the value of $e^{y(\frac{\pi}{6})}$ is equal to ...........

Let $y=y(x)$ be the solution of the differential equation $e^{x} \sqrt{1-y^{2}} dx + \frac{y}{x} dy = 0$,with $y(1) = -1$. Then the value of $(y(3))^{2}$ is equal to:

The solution of the differential equation $x = 1 + xy\frac{dy}{dx} + \frac{(xy)^2}{2!}\left(\frac{dy}{dx}\right)^2 + \frac{(xy)^3}{3!}\left(\frac{dy}{dx}\right)^3 + \dots$ is

The general solution of $\frac{dy}{dx} = \frac{x+y+1}{x+y-1}$ is