Wednesday, 23 August 2017

SOLUTION OF LINEAR THIRD ORDER NON-HOMOGENEOUS ORDINARY DIFFERENTIAL EQUATION USING MATRIX METHOD



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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