**Studying Different Numerical Methods in Solving First Order Differential Equations**

**CHAPTER ONE**

**1.0 INTRODUCTION**

**1.1 BACKGROUND OF STUDY**

Differential equations can describe nearly all systems changing. They are ubiquitous in science and engineering as well as economics, social science, biology, business, health care, etc. Many mathematicians have studied the nature of these equations for hundreds of years and there are many well-developed solution techniques. Often, systems described by differential equations are so complex, or the systems that they describe are so large, that a purely analytical solution to the equations is not tractable. It is in these complex systems where computer simulations and numerical methods are useful. The techniques for solving differential equations based on numerical approximations were developed before programmable computers existed. During World War II, it was common to find rooms of people (usually women) working on mechanical calculators to numerically solve systems of differential equations for military calculations. Before programmable computers, it was also common to exploit analogies to electrical systems to design analog computers to study mechanical, thermal, or chemical systems. As programmable computers have increased in speed and decreased in cost, increasingly complex systems of differential equations can be solved with simple programs written to run on a common PC. Currently, the computer on your desk can tackle problems that were inaccessible to the fastest supercomputers just 5 or 10 years ago.

The standard initial value problem is to determine a vector-valued function: y: [t0, T] Rd with a given initial value y(t0) = y0 є Rd such that the derivative y1(t) depends on the current solution value y(t) for every at every t є [t0,T] in a prescribed way y1(t) = f(t,y(t)) for t0≤t≤T, y(t0) = y0

Here, the given function f is defined on an open subset of R×Rd containing (t0, y0) and takes values in Rd. If f is continuously differentiable then there exists a unique solution at least locally on some open interval containing t0. In many applications, t represents time, and it will be convenient to refer to t as time in what follows.

**1.2 STATEMENT OF PROBLEM**

The study will try to solve the problems below using different types of numerical methods

y1 = y2 +1, y(0)= 0 on the interval [0,1] with h=0.1

y1 = y-x, y(0) = 2 on the interval [0,1] with h=0.1

**1.3 AIM AND OBJECTIVES OF THE STUDY**

The main aim of the research work is to examine different numerical methods in solving first-order differential equations. Other specific objectives of the study are:

To determine the relationship between the numerical methods of solving first-order differential equations.

To investigate the factors affecting the numerical methods for solving differential equations.

To determine the convergence of these numerical methods.

**1.4 RESEARCH QUESTIONS**

The study came up with research questions to ascertain the above-stated objectives of the study. The research questions for the study are:

What is the relationship between the numerical methods of solving first-order differential equations?

What are the factors affecting the numerical methods for solving differential equations?

What is the convergence of these numerical methods?

**1.5 SIGNIFICANCE OF STUDY**

The study on different numerical methods in solving first-order differential equations will be of immense benefit to the mathematics department in the sense that the study will solve first-order differential equations using different numerical methods. The findings of the study will serve as a repository of information to other researchers that desire to carry out similar research on the above topic. Finally, the study will contribute to the body of existing literature and knowledge in this field of study and provide a basis for further research

**1.6 SCOPE OF STUDY**

The study on different numerical methods in solving first-order differential equations will focus on the Runge-Kutta method and adam Bashford method. The study will also cover the solution of the first-order differential equation using the Runge-Kutta method and adam Bashford method.

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