The differential equation $\cot y \, dx = x \, dy$ has a solution of the form

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

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

The slope of a curve at any point is inversely proportional to twice the $y$-coordinate of that point. If the curve passes through $(4, 3)$,then the equation of the curve is:

Given that the slope of the tangent to a curve $y=y(x)$ at any point $(x, y)$ is $\frac{2y}{x^2}$. If the curve passes through the centre of the circle $x^2+y^2-2x-2y=0$,then its equation is

The particular solution of the differential equation $(1+y^2)(1+\log x) dx + x dy = 0$ at $x=1, y=1$ is