This is a problem in "Differential Equations: A Modern Approach with Wavelets" by Krantz:
If the length of any small portion of an elastic cable of uniform density is proportional to the tension in it, then show that it assumes the shape of a parabola when hanging under its own weight.
After not knowing how to approach the problem, I decided to research it and I ended up finding about the elastic catenary. And another source says that it is a non-algebraic curve, so definitely not a parabola.
Is the problem wrong? or is it that the problem is not that of the elastic catenary?