The differential equation of all lines perpendicular to the line $5x + 2y + 7 = 0$ is

Statement $I$: The differential equation corresponding to the family of circles having their centres on $Y$-axis and fixed radius $k$ is $(x^2-k^2)(\frac{dy}{dx})^2+x^2=0$.
Statement $II$: The differential equation corresponding to the family of circles passing through the origin and having their centres on $X$-axis is $x^2-y^2+2xy \frac{dy}{dx}=0$.
Which of the above statements is (are) true?

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

The differential equation of the family of all parabolas whose axis is the $y$-axis is ...

The differential equation obtained by eliminating the arbitrary constant from the equation $y^2 = (x + c)^3$ is

If $m$ and $n$ are the order and degree of the differential equation of the family of parabolas with focus at the origin and $X$-axis as its axis, then $m n-m+n=$