The differential equation for which $y = ax^2 + bx + c$ is the general solution is:

If the order of a differential equation $\frac{d^2 y}{d x^2}-2\left(\frac{d y}{d x}\right)^3+\sin \left(\frac{d y}{d x}\right)+y=0$ is $l$ and the degree of the differential equation $\left(1+\frac{d^2 y}{d x^2}\right)^{\frac{2}{3}}=\left[2-\left(\frac{d y}{d x}\right)^3\right]^{\frac{3}{2}}$ is $m$, then the differential equation corresponding to the family of curves $y=A x^l+B e^{m x}$, where $A$ and $B$ are arbitrary constants, is

Form the differential equation of the family of parabolas having vertex at origin and axis along the positive $y$-axis.

The differential equation whose solution is $(x - h)^2 + (y - k)^2 = a^2$ is ($a$ is a constant).

Form the differential equation representing the family of ellipses having foci on the $x$-axis and centre at the origin.

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