Although I'm not a professional mathematician, I'm frequently involved in tutoring. I was asked by a student to find a curve in which the area in square units between the curve and the X axis would be equal to the length of said curve in linear units. The problem has a trivial solution of y=1 or y=-1. It's also well known that the curve y=cosh x describes such a curve.
A complete solution would of course be a solution to the differential equation Y^2 = 1 + (dy/dx)^2. How does one solve this equation?