The differential equation of all parabolas,whose axes are parallel to the $Y$-axis,is

The differential equation of the family of circles passing through the points $(0,2)$ and $(0,-2)$ is

Let $c_1, c_2, c_3, c_4$ be arbitrary constants. The order of the differential equation,corresponding to $y=c_1 e^x+c_2 e^{\log _{e} x}+c_3 \sin ^2 x-c_4\left(\cos ^2 x-1\right)$ is

$y = \frac{x}{x + 1}$ is a solution of which differential equation?

Verify that the given function $xy = \log y + C$ is a solution of the differential equation $y' = \frac{y^2}{1 - xy}$ $(xy \neq 1)$.

The differential equation of the family of ellipses $\frac{x^2}{a^2} + \frac{y^2}{b^2} = c$ is given by $\left( y' = \frac{dy}{dx}, y'' = \frac{d^2y}{dx^2} \right)$.