Let $p \in \mathbb{R}$. Then the differential equation of the family of curves $y=(\alpha+\beta x) e^{p x}$, where $\alpha$ and $\beta$ are arbitrary constants, is

Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.

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

Verify that the given function $y=ae^{x}+be^{-x}+x^{2}$ is a solution of the differential equation $x \frac{d^{2} y}{dx^{2}}+2 \frac{dy}{dx}-xy+x^{2}-2=0$.

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 all the lines in the $xy$-plane is