$y = \frac{x}{x + 1}$ is a solution of which differential equation?

The differential equation for which $y = ax^2 + bx + c$ is the general solution is:

The differential equation whose solution is $y=e^{ax}$ is

The differential equation of the family of parabolas with vertex at $(0,-1)$ and having axis along the $Y$-axis is

$y = ke^{\sin ^{-1} x} + 3$ is a solution of which differential equation?

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.