Rough estimates and binomial approximations for Crocco's equation in the boundary problems

In order to solve the Crocco boundary problems known as the typical one and the uniform one, binomials (as approximants of exact solutions) and integral identities have been used. The extent of the closeness of the exact solution to its approximation was estimated using the φ(0) value. The solution of the typical Crocco boundary problem was proved to have a logarithmic singularity of the derivative at φ = 0. Crocco's equation was found to provide both necessary and sufficient conditions for the minimum of a positive distribution being vortex in dφ/dh. The uniform Crocco boundary problem was demonstrated to be equivalent to the two typical Crocco boundary problems with a common critical point.