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

The differential equation formed by eliminating arbitrary constants $A$ and $B$ from the equation $y = A \cos 3x + B \sin 3x$ is

The differential equation representing the family of ellipses having foci either on the $x$-axis or on the $y$-axis,centered at the origin,and passing through the point $(0,3)$ is:

The degree of the differential equation whose solution is $y^2=8a(x+a)$ is

Which of the following differential equations has $y=c_{1} e^{x}+c_{2} e^{-x}$ as the general solution?

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 $\frac{df}{dx}$.