ANALYSIS AND COMPARATIVE STUDY OF NUMERICAL METHODS TO SOLVE ORDINARY DIFFERENTIAL EQUATION WITH INITIAL VALUE PROBLEM.

Since mathematics is a science of communication between us and the scientific sciences, in particular it introduces all the rules and problems as formulas, and searches for a solution. A part of the mathematics that is widely used in all sciences is the differential equation that will be studied in this dissertation. Each parts of these equations has its own method for solving, and we have generally studied the analytic methods in the calculus, and here we will introduce the numerical solutions. It is worth noting that in analytic methods cannot gives solution, for all equations this is where scientists have discovered the numerical methods that can be solved by those methods for those equations that are not solved in an analytical methods.In this Article, we first introduce differential equations and introduce a number of elementary topics for introduction so that the reader will get acquainted with these definitions and issues before the start of the process.Later in this article, one basic methods will be studied, the single step method, respectively, which relate to the initial value problem.Here we will examine in detail and analyze all the ways in which these methods are available.In the next step, all the good nesses, advantages, disadvantages that exist between these methodswill be discussed, and also a comparative study we will have in this paper.