**CHAPTER ONE**

**1.1**

**INTRODUCTION**

Generally, differential equations
arises from a series of practical science and engineering problems. Any
equations connecting a variable x, some functions of x, f(x), and certain
differentials of x is called a differential equation.

Differential equation are
classified into two different types, namely;

**i.**Ordinary Differential Equation (O.D.E) and

**ii.**Partial Differential Equation (P.D.E).

A differential equation
containing an independent variable x and total derivatives only is called an
Ordinary differential equation. Also, if a differential equation contains
Partial derivatives, such differential equation is called a Partial
differential equation (P.D.E).

A differential equation is said
to be linear if the unknown function and its derivatives appears to have a
power of one, we have to note here that products are not allowed.

Differential equations have so
far been well recognized in engineering, economics and other disciplines
because of its great applications in solving a deterministic relation involving
some continuously varying quantities ( using functions) and their rates of
change in space or time as their
derivatives is established.

Matheatically, differential
equations are studied from several perspectives, mostly concerned withns, also
we have their solutions, some of them
are; the analytical method which puts emphasis on qualitative analysis of
systems described by different equations, also we have the numerical methods
which has to do with how to determine solutions with a given degree of
accuracy.

Differential equation a very
important field of study and its being treated widely in pure and applied
mathematics, engineering, and Physics. Because all of these disciplines are
concerned with the many properties of differential equations of various types.
Existence and uniqueness of solutions are majorly considered in pure
mathematics, while applied mathematics deals with the adequate and proper
justification of the methods for approximating solutions. Differential equations
are also applicable in solving real-life problems, in this case the solutions
can be approximated using numerical methods because the problems may not be
directly solvable. Many important laws in science, like physics and chemistry,
can be used to form a differential equations.

One other importance of
differential equation is its usefulness in modeling the behavior of difficulty
systems in biology and economics.

The principal of differential
equation is well developed in such a way that different methods can be used to
solve them which varies significantly with the type of the equation. Linear systems are best study solved by the
use of matrices of which, however, only modest knowledge will be needed here.

Matrix is the latin word for
womb, and it means that sense in English. Generally, matrix means any place in
which something is formed or produced. Matrix in mathematics means rectangular
array of numbers, symbols, or expressions arranged in rows and columns.

The items in a matrix are called
its elements or entries.

Matrices of the same size can be
added or subtracted. The horizontal and vertical line in a matrix are called
rows and columns respectively. To Specify the size of a matrix, a matrix with
m-rows and n-colums is called an m-by-n matrix or mxn matrix, while m and
n its dimensions.

**1.2**

**STATEMENT OF PROBLEM**

Solution of non-homogeneous
linear differential equation is rarely touched into etail and because of this
reason, many students see it as a difficult aspect of mathematics.

This research work is focused on
solving non-homogeneous linear third order ordinary differential equation with
the method of matrix.

**1.3**

**AIMS OF THE STUDY**

The
aims of this research work are as follows;

·
To show the procedures in applying
matrix method in the solution of non
homogeneous linear third order differential equation.

·
Solves problems involving linear third
order ordinary differential equations using matrix method.

·
Explain how differential equations can
serve as a problem solving tool in real life situation.

**1.4 OBJECTIVE OF THE STUDY**

This
research work is brought-up in order to provide and understanding of
non-homogeneous linear third order differential equations and to give method
for solving them.

The objectives of this study are;

·
Solve applied problems that involves
linear third order differential equations.

·
Find the solution of a third order
linear differential equations.

**1.4**

**LIMITATION AND SCOPE OF STUDY**

This
work is focused on solving non-homogeneous linear third order differential
equation using matrix method. Conversion of differential equation into matrix
form and applying the method of interest to solving the equations shall be
considered.

The
limitation encountered in carrying out this research work is the lack of power
supply and materials in terms of textbooks.

**1.5**

**LAYOUT OF THE WORK**

This
research work did not cover in details all about the solution of
non-homogeneous linear third order ordinary differential equations but it will
serve as a background knowledge for anyone who is willing to go further in the
subject of study.

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