Numerical method for nonlinear differential equation with arbitrary BC

The aim of this work consists in developing an algorithm and a numerical calculation program allowing to solving a nonlinear arbitrary 2nd and 3rd order differential equation (DE) of with generalized Cauchy boundary conditions (BC) over the interval [a1, a2]. The Dirichlet and Neumann BC becomes a particular case. The problem consists in transforming the DE to a system of n(n+1) nonlinear DE of the first order (FODE) with initial n conditions (IC), of which n equations justify the function y(x) and these (n-1) successive derivatives, and of n2 functions again witch justify the transformation of the DE towards a system of FODE with IC. The number n is the order of the DE. The resolution of this system of equations is made by the adaptation of the Runge Kutta method of order 4. The determination of the IC is made by the resolution of an algebraic system of n nonlinear equations, of which the resolution is made simultaneously by the Newton's method. For each iteration of Newton¿s method, a system of nonlinear algebraic equations is obtained whose solution is made by the method of Gauss.