## Wednesday, 23 August 2017

### SOLUTION OF LINEAR ORDINARY DIFFERENTIAL EQUATION USING FINITE DIFFERENCE METHOD

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
Y2 = 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
Y2 =2y.dy/dx.x                                                                                     (1.3)
Y =2xdy/dx                                                                                         (1.4)
Y2 -2xy=0                                                                                             (1.5)
ELIMINATION OF ARBITRARY FUNCTIONS
Given the function
Z = f(x2+y2)                                                                                                                (1.6)
Where  = f(x2+y2) is the arbitrary function to be eliminated.
To do this, we normally replace the argument x2 + y2 by U
So that U= x2 + y2
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 nth 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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