The solution of the differential equation $\frac{dy}{dx} = 2xy$ is

If $\frac{dy}{dx} + \frac{1}{\sqrt{1 - x^2}} = 0$,then

The general solution of the differential equation ${e^y}\frac{{dy}}{{dx}} + ({e^y} + 1)\cot x = 0$ is

$A$ curve having the condition that the slope of the tangent at any point $(x, y)$ is two times the slope of the straight line joining the same point to the origin $(0, 0)$ is a/an:

The solution of the equation $(2y - 1) \, dx - (2x + 3) \, dy = 0$ is

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