The Numerical Solution of the Cauchy Problem for Ordinary Differential Equations

Abstract

In the present chapter we will deal with numerical methods for solution of ordinary differential equations or of systems of such equations. To be definite, we shall speak now about one differential equation of order n $${y^{\left( n \right)}} = f\left( {x,y,y' \ldots ,{y^{\left( {n - 1} \right)}}} \right).$$ (1.1) It is known that the solution of equation (1.1) is not determined uniquely by this equation. The general solution of this equation depends, generally speaking, on n arbitrary constants. Numerical methods are applicable to finding particular solutions of the equation (1.1). In order to obtain this particular solution of the differential equation (1.1), we have to impose some n additional conditions upon the sought for solution.