>>13434563We have a hanging chain with a weight-per-unit-length of w0, where the lowest point of the chain passes (horizontally) through the origin. The tension at the origin is T0 while the tension at a point (x,y) is tangent to the chain and equal to T. Let the arc length from the origin to (x,y) be s, and the slope of the tangent with respect to the horizontal be . Then, doing the force balance for that segment, we get and . DEFINE . Divide both previous equations to cancel out tension at the point, so in the infinitesimal limit. Differentiate to get but from basic calculus. Let then . Solve the differential equation to get and once more to finally reach
Finally, the height at x=0 is y(0)=a which proves that is, in fact, the height of the lowest point of the chain above the origin.
You said you understood the derivation, anon.
