The differential equation of which $xy = ae^x + be^{-x} + x^2$ is a solution,is

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

If $m$ and $n$ are respectively the order and the degree of the differential equation representing the family of curves $y^2 - 5ax - 5a^{3/2} = 0$ ($a > 0$ is a parameter),then the value of $m - n$ is

The differential equation of the family of curves $y=e^{x}(A \cos x+B \sin x)$,where $A$ and $B$ are arbitrary constants,is

Statement $(I)$: The elimination of arbitrary constants $\alpha, \beta$ and $\gamma$ from $y=(\alpha+\beta+\gamma) x$ results in a differential equation of order three.
Statement $(II)$: The elimination of arbitrary constants $\alpha, \beta$ and $\gamma$ from $y=\alpha x+\beta \sin x+\gamma e^x$ results in a differential equation of order three.

The differential equation of the family of circles passing through the origin and having their centres on the $x$-axis is