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

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 whose general solution is given by $y = (c_1 \cos(x + c_2)) - (c_3 e^{(-x + c_4)}) + (c_5 \sin x)$,where $c_1, c_2, c_3, c_4, c_5$ are arbitrary constants,is

The differential equation corresponding to the family of circles given by $(x-a)^2+(y-b)^2=4$,where $a$ and $b$ are parameters,is

The differential equation for which $y^2 = 4a(x + a)$ (where $a$ is a parameter) is the general solution, is

The differential equation of the family of all straight lines passing through the origin is