Bernoulli's Hanging Chain Problem

Theorem
Consider a uniform chain $C$ whose physical properties are as follows:
 * $C$ is of length $l$
 * The mass per unit length of $C$ is $m$
 * $C$ is of zero stiffness.

Let $C$ be suspended in a vertical line from a fixed point and otherwise free to move.

Let $C$ be slightly disturbed in a vertical plane from its position of stable equilibrium.

Let $y \left({t}\right)$ be the horizontal displacement at time $t$ from its position of stable equilibrium of a particle of $C$ which is a vertical distance $x$ from its point of attachment.

The $2$nd order ODE describing the motion of $y$ is:
 * $\dfrac {\mathrm d^2 y} {\mathrm d t^2} = g \left({l - x}\right) \dfrac {\mathrm d^2 y} {\mathrm d x^2} - g\dfrac {\mathrm d y} {\mathrm d x}$

Proof
He solved this problem around the year 1732.