If the differential equation representing the family of all circles touching the $x-$axis at the origin is $(x^2 - y^2)\frac{dy}{dx} = g(x)y$,then $g(x)$ equals

The order of the differential equation whose solution is $a e^{x} + b e^{2x} + c e^{3x} + d = 0$ is:

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

The differential equation for which $\sin^{-1} x + \sin^{-1} y = c$ is given by

If $a$ and $b$ are arbitrary constants,then the differential equation having $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ as its general solution is

If $\alpha$ and $\beta$ are respectively the order and degree of the differential equation for which $a x^2+b y^2=1$ is the general solution, then the eccentricity of the ellipse $\alpha x^2+\beta y^2=1$ is