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 $\map y t$ 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:
 * $\dfrac {\d^2 y} {\d t^2} = g \paren {l - x} \dfrac {\d^2 y} {\d x^2} - g \dfrac {\d y} {\d x}$

Historical Note
This problem was solved by around the year $1732$.