The differential equation of $3y = \sqrt[3]{x} + c$ is

The equation which represents the system of parabolas whose axis is parallel to the $y$-axis satisfies the differential equation:

Find the differential equation for the family of curves given by $y = a e^{3x} + b e^{-2x}$ by eliminating the arbitrary constants $a$ and $b$.

The general solution of the differential equation of all circles having center at $A(-1, 2)$ is $ . . . . . . $.

If the family of curves $y = a e^{4x} + b e^{-x}$, where $a, b$ are arbitrary constants, represents the general solution of the differential equation $f(x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}) = 0$, then find $\frac{df}{dx}$.

The differential equation of the simple harmonic motion given by $x=A \cos (n t+\alpha)$ is