24.1. INTRODUCTION

An ordinary differential equation of n ^th order is of the form

The general solution of this equation contains n arbitrary constants or parameters. Thus the general solution is of the form f( x, y, c ₁, c ₂, ......., c _n) = 0.

If particular values are given to the constants c ₁, c ₂, ......, c _n, then the resulting solution is called a particular solution. To get a particular solution we must be given n conditions ( e.g., the values of y or its derivatives for some specific values of x) to determine the n constants. If all the n conditions are specified at the same initial point x ₀ then the problem is called an initial value problem. When the conditions are specified at two or more values of x, then the problem is called a boundary value problem.

Many ordinary differential equations can be solved by analytical methods discussed earlier giving closed form solutions i.e., expressing y in terms of a finite number of elementary functions of x. However, a majority of differential equations appearing in physical problems cannot be solved analytically. Thus it becomes imperative to discuss their solution by numerical methods.

In this chapter, we shall discuss some of the methods for obtaining numerical solutions of first-order and first-degree ordinary differential equations subject to the condition y( x ₀) =