**CHAPTER ONE**

**1.0 INTRODUCTION**

Differential equation is a relation
connecting the function y and an independent variable x and its derivatives of
various orders. It is also an equation that involves derivatives. Differential
equations differ from ordinary equations of mathematics in that, in addition to
variables and constants they also contain derivatives of one or more of the
variables involved.

Differential equations are categorized
into two major categories; ordinary differential equation (ODE), in which the
unknown functions (also known as the dependent variables) is a function of a
single independent variable; partial differential equation (PDE) in which the
unknown function is the function of multiple independent variables and the
equation involves it partial derivatives. A differential equation is linear if
the unknown function and it derivatives appear to the power of one (product are
not allowed).

Historically, differential equations
have originated in chemistry, physics and engineering, more recently the have
also arisen in models medicine, biology, anthropology and many more science and
technology discipline.

Differential equations are studied from
several different perspectives, mostly concerned with their solutions, the set
of functions that satisfy the equation. The theory of dynamical systems put
emphasis on qualitative analysis of systems described by differential equation,
while many numerical methods have been developed to determine solution with a
given degree of accuracy.

The theory of differential equation is
well developed and the methods used to study vary significantly with the type
of equation. Second order differential equations are best treated by the use of
finite difference method which we intend to use in this project work.

**FORMATION OF ORDINARY DIFFERENTIAL EQUATION**

Formulation of ordinary differential
can be done in two different ways, namely:

i.
Method
of elimination of arbitrary constants.

ii.
Method
of elimination of arbitrary functions.

**ELIMINATION OF ARBITRARY CONSTANTS**

Given a relation of the form

Y

^{2}= 4ax (1.1)
Where

**a**is arbitrary constants
X is independent variable

Y is dependent variable

Differentiating with respect to X

We get 2ydy/dx=4a
(1.2)

Substituting 4a in (1.1) we get

Y

^{2}=2y.dy/dx.x (1.3)
Y =2xdy/dx
(1.4)

Y

^{2}-2xy=0 (1.5)**ELIMINATION OF ARBITRARY FUNCTIONS**

Given the function

Z = f(x

^{2}+y^{2}) (1.6)
Where = f(x

^{2}+y^{2}) is the arbitrary function to be eliminated.
To do this, we normally replace the
argument x

^{2}+ y^{2 }by U
So that U= x

^{2}+ y^{2}
We have Z = f (u) (1.7)

dz/dx=p=f1(u)du/dx
(1.8)

dz/dy=q=f1(u)du/dy
(1.9)

This eliminate f completely to get

xq
–yp = 0 (1.10)

**1.2 STATEMENT OF PROBLEM**

The use of finite difference method in
solving second order differential equation is not commonly approached in
details and because of this, many student have found it as a very difficult
aspect of mathematics. This research work is mainly concern with how to use
finite difference method to solve second order differential equation.

**1.3 AIMS OF THE STUDY**

The aims of this research work are as
follow:

1. To introduce the reader to the basic
aspects of the subject

2. To solve problems involving second
order differential equation using finite difference method.

3. To explain the usefulness of
differential equation as a problem solving tool in real life situation.

4. To analyze the steps in the application
of finite difference method in solving second order differential equation.

**1.4 OBJECTIVES OF THE STUDY**

1. The general purpose of this project
work is to provide an understanding of second-order differential equation and
to give method of solving them

2. Solve appropriate differential
equation with finite difference method.

**1.5 LIMITATION OF THE STUDY**

This project work will be restricted to
the application of finite difference method in solving second order
differential equation. Inorder to achieve this, this project work will be
treated in four comprehensive chapters. The limitation encountered in carrying
out this research work was the lack of sufficient research reference materials.

**1.6 DEFINITION OF TERMS**

**Equation:**Equations are used to describe the relationship between the dependent variables and independent variables.

**Differential Equation:**Equation that involve dependent variables and their derivatives with respect to the independent variables are called differential equations.

**Ordinary Differential Equation:**Differential equation that involve only one independent variable are called ordinary differential equation.

**Partial Differential Equation:**Differential equation that involve two or more independent variables are called partial differential equation.

**Initial Condition:**Constraint that are specified at the initial point, generally time point are called initial condition. Problems with specified initial conditions are called initial value problems.

**Boundary Condition:**Constraints that are specified at the boundary points, generally space points are called boundary condition problem. Problems with specified boundary conditions are called BVP.

**Degree of Differential Equation:**The degree of a differential equation is the power of the highest derivative term.

**Linear Differential Equation:**A differential equation is called linear if there are no multiplication among dependent variables and their derivatives. In other words, all coefficients are functions of independent variables.

**Non-Linear Differential Equation:**Differential equations that do not satisfy the definition of linear are non-linear.

**Homogeneous Differential Equation:**A differential equation is homogeneous if every single term contains the dependent variables or their derivatives. A differential equation whose right hand side is equal to zero is known as a homogeneous differential equation.

**Non-Homogeneous Differential Equation:**Differential equations which do not satisfy the definition of homogeneous are referred to be non-homogeneous.

**General Solution:**Solutions obtained from integrating the differential equations are called general solutions. The general solutions of a n

^{th}order ordinary differential equation contains

^{n }arbitrary

**constraints resulting from integrating**

^{n }times.

**Particular Solution:**Particular solutions are the solution obtained by assigning specific values to the arbitrary constraints in the general solutions.

**Singular Solution:**Solutions that cannot be expressed by the general solution are called singular solutions.

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