The order of the differential equation of the family of all concentric circles centered at $(h, k)$ is

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

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 for all the straight lines which are at a unit distance from the origin is

Verify that the given function $y = x^{2} + 2x + C$ is a solution of the differential equation $y' - 2x - 2 = 0$.

The differential equation of the circles having their centres on the line $y=8$ and touching the $X$-axis is