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

The differential equation of $y=e^x(a \cos x+b \sin x)$ is

The normal form of the equation of a line is $x \cos \alpha + y \sin \alpha = p$. Find the differential equation of the family of all such lines, where $p$ and $\alpha$ are arbitrary constants.

Let $f(x)$ be a positive function such that the area bounded by $y=f(x), y=0$ from $x=0$ to $x=a>0$ is $e^{-a}+4a^2+a-1$. Then the differential equation,whose general solution is $y=c_1 f(x)+c_2$,where $c_1$ and $c_2$ are arbitrary constants,is :

$A$ differential equation representing the family of parabolas with axis parallel to the $y$-axis and whose length of latus rectum is the distance of the point $(2, -3)$ from the line $3x + 4y = 5$,is given by:

The curve $y=(\cos x+y)^{1 / 2}$ satisfies the differential equation