PROJECT
BY
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EE/2017/167
SUBMITTED TO
DEPARTMENT OF ELECTRICAL ELECTRONIC ENGINEERING FACULTY OF ENGINEERING CARITAS UNIVERSITY, AMORJI-NIKE, ENUGU.
IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE AWARD OF BACHELOR OF ENGINEERING (B.ENG)
APPROVAL PAGE
This project has been read and approved by the undersigned as with the requirement at the department of Electrical Electronic Engineering of Caritas University Amorji Nike Enugu for the award of Bachelor of Engineering (B.Eng.) in Electrical Electronic Engineering.
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Engr. Ejimorfor Date
(Project supervisor)
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Engr. Ejimofor Date
(Head of Department)
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External Supervisor Date
DECLARATION
I declare that this project material is an original work done by me under the supervision of Engr. Ejimorfor, department of electrical electronic engineering faculty of engineering caritas university, amorji-nike, enugu
DEDICATION
This project is dedicated to Almighty God and to my parents Engr. & Mrs. Chukwu A. Orioha and to my beloved brothers and sisters whose ever loving kindness and support has seen me through my years of studies.
ACKNOWLEDGEMENT
I wish to express my immense gratitude to God Almighty for his mercy, guidance and protection towards me for seeing me through the rigors of this work. I am greatly indebted to my supervisor Engr. Ejimofor for his kind gesture and whose critics lead to the achievement of this work. I also will remain grateful to the tremendous contribution of my lecturers Engr. Ojobor (the Dean of Engineering Faculty), Engr. Ejimofor (Head of Electrical Electronic Engineering Department), Engr. Mbah, Engr. Ochi, and all the staff of Electrical Electronic Engineering both academic and non academic staff for their intellectual upbringing. My special appreciation goes to my loving parents Engr. & Mrs. Chukwu A. Orioha, my grandparent, my uncles and aunties, my brothers and sisters whose moral and financial support cannot be over emphasized. Also my sincere gratitude and special regards to my friends too many to mention whose encouragement also lead to the success of this work.
an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and its derivatives. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable. The few non-linear ODEs that can be solved explicitly are generally solved by transforming the equation into an equivalent linear ODE
Some ODEs may be solved explicitly in terms of known functions and integrals. When it is not possible, one may often use the equation for computing the Taylor series of the solutions. For applied problems, one generally use numerical methods for ordinary differential equations for getting an approximation of the desired solution.
CHAPTER ONE
1.1 INTRODUCTION
A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions—the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form.
If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy. In mathematics, there are three different types of differential equation which are:
However, in this work an ordinary differential equation (ODE) is studied which is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x. The unknown function is generally represented by a variable (often denoted y), which, therefore, depends on x. Thus x is often called the independent variable of the equation. The term "ordinary" is used in contrast with the term partial differential equation, which may be with respect to more than one independent variable.
1.2 BACKGROUND OF THE STUDY
Ordinary differential equations (ODEs) arise in many contexts of mathematics and science (social as well as natural). Mathematical descriptions of change use differentials and derivatives. Various differentials, derivatives, and functions become related to each other via equations, and thus a differential equation is a result that describes dynamically changing phenomena, evolution, and variation. Often, quantities are defined as the rate of change of other quantities (for example, derivatives of displacement with respect to time), or gradients of quantities, which is how they enter differential equations.
Specific mathematical fields include geometry and analytical mechanics. Scientific fields include much of physics and astronomy (celestial mechanics), meteorology (weather modelling), chemistry (reaction rates),[2] biology (infectious diseases, genetic variation), ecology and population modelling (population competition), economics (stock trends, interest rates and the market equilibrium price changes).
Many mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, d'Alembert, and Euler.
1.3 OBJECTIVE OF THE STUDY
The objective of this work is have a series solution to ordinary differential equation with different formulas.
1.4 APPLICATIONS OF THE STUDY
The study of differential equations is a wide field in pure and applied mathematics, physics, and engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modelling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods.
Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider the propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order partial differential equation, the wave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by Joseph Fourier, is governed by another second-order partial differential equation, the heat equation. It turns out that many diffusion processes, while seemingly different, are described by the same equation; the Black–Scholes equation in finance is, for instance, related to the heat equation.
1.5 SCOPE OF THE STUDY
In this introductory course on Ordinary Differential Equations, we first provide basic terminologies on the theory of differential equations and then proceed to methods of solving various types of ordinary differential equations. We handle first order differential equations and then second order linear differential equations. We also discuss some related concrete mathematical modeling problems, which can be handled by the methods introduced in this course.
1.6 PURPOSE OF THE STUDY
This course is recommended for undergraduate students in mathematics, physics, engineering and the social sciences who want to learn basic concepts and ideas of ordinary differential equations. Learners are required to know usual college level calculus including differential and integral calculus.
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