The optimum shape of an hydrofoil with no cavitation

We consider a two-dimensional hydrofoil at rest in the (xy)-plane embedded in a steam with a uniform flow at infinity and we pose the problem of finding the optimum shape of the hydrofoil of a given length and prescribed mean curvature for which the lift is a maximum. Using the lifting line theory and standard variational calculus techniques we show that the slope of the mean chord of the hydrofoil has to satisfy a differential equation of the second order. The Rayleigh-Ritz method is used to solve the second order differential equation which gives the optimal values.