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SECOND ORDER DIFFERENTIAL EQUATIONS, METHOD OF SOLUTIONS AND APPLICATIONS

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 Format: MS WORD ::   Chapters: 1-5 ::   Pages: 57 ::   Attributes: problem and solution ::   151 people found this useful
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CHAPTER ONE

INTRODUCTION

1.1 BACKGROUND OF THE STUDY

Before embarking on a serious study of differential equations (for example, by reading this project) one should have some idea of the possible benefits to be gained by doing so. For some people the intrinsic interest of the subject itself is enough motivation, but for most it is the likelihood of important applications to other fields that makes the undertaking worthwhile. Many of the principles, or laws, underlying the behaviour of the natural world are statements or relations involving rates at which things happen. When expressed in mathematical terms the resulting relations are differential equations, that is, equations containing derivatives. Therefore, to understand and investigate problems involving the motion of fluids, the flow of current in electric circuits, the dissipation of heat in solid objects, the propagation and detection of seismic waves, or the increase or decrease of populations, among many others, it is necessary to know something about differential equations. A differential equation that describes some physical process is often called a mathematical model of the process, and five of such models are discussed in this project. In applying differential equations to any of the numerous fields in which they are useful, it is necessary first to formulate the appropriate differential equation that describes, or models, the problem being investigated. In this project we will focus more on ordinary second order linear differential equations which can be used to model real life problems in various aspects such as: Physics, Biology, Finances, Engineering, just to name a few. Many interesting ordinary differential equations (ODEs) arise from applications. One reason for understanding these applications is that you can combine your physical intuition with your mathematical intuition in the same problem, and the result is usually an improvement on both. Given a differential equation

p(x)y’’ + q(x)y’ + r(x)y = g(x), y(x0) = y0, y0 (x0) = y’’ , (1.1)

where p, q, r, and g are given continuous functions in some interval a < x < b containing the points x0 and y0, y’’ , which are arbitrary prescribed initial values. Any differentiable function y = ψ(x), satisfying Eq.(1.1) for all x in some interval (a, b) is called the solution of Eq.(1.1) and our objective is to determine whether such functions exist. Unfortunately, for arbitrary functions p, q, r, g, there is no general method for solving the equation in terms of elementary functions. Before we look at how to solve this problem, we need to address the following question: ”What constitute a properly formulated problem? ”This question will be answered by the theorem we are going to state and prove subsequently. The importance of that theorem lies in the fact that, it defines the framework within which we can construct solutions.

1.2 STATEMENT OF THE PROBLEM

The study will try to solve the problems below:

y″ -xy′ + 2y = 0

and x2y″ + y′ + xy = 0 using laplace transform, variations of parameter and the method of undertermine coefficient

1.3 AIM AND OBJECTIVES OF THE STUDY

The study seeks to examine the second order differential equation, methods of solutions and applications. The objectives of the study are:

  1. To determine the various types of second order differential equations
  2. To identify the various methods for solving homogenous and non-homogenous secondary order differential equation
  3. To determine the variation in effeciency of the application of various methods of solutions of second order differential equation
  4. To identify the factors affecting the second order differential equation, methods of solutions and applications

1.4 SIGNIFICANCE OF THE STUDY

This study will be of immense benefit to other researchers who intend to know more on this topic and can also be used by non-researchers to build more on their work.

This study contributes to knowledge and could serve as a bench mark or guide for other work or study. It avails individuals with information about the second order differential equation, methods of solutions and applications.

1.5 SCOPE OF THE STUDY

The study covers on the second order differential equation, methods of solutions and applications with focus on laplace transforms, variation of parameters and the method of undertermine coefficient

1.6 DEFINITION OF TERMS

 ODE: differential equation

Differential equation: A differential equation is a mathematical equation that relates some function with its derivatives

 


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Project Information

Format:MS WORD
Chapter:1-5
Pages:57
Attribute:problem and solution
Price:₦3,000
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