Hanging cable

A cable is suspended between two towers like a power line between two towers. Assume that the towers are on a level parcel of ground but the heights of the towers maybe different. Using the following notation

$$ u(x) $$ - height of cable center

$$ L $$ - distance between two towers

$$ u_{0} $$ - height of cable at tower 1

$$ u_{L} $$ - height of cable at tower 2.

$$ T $$ - horizontal component of tension.

$$ w $$ units of weight per unit length of cable.

and the forces acting on the cable, it can be shown that the cable height $$ u(x) $$ is the solution of the differential equation

$$ {d^{2} u(x) \over {dx^{2}}} = {w \over T} \sqrt{1 + ( {du \over {dx}} ) ^ {2} } $$

with the end conditions $$ u(0)= u_{0} $$ and $$ u(L)=u_{L} $$.