THE RELEVANCE OF THE USE OF METHOD OF UNDETERMINE COEFFICIENTS FOR SOLVING DIFFERENTIAL EQUATIONS

THE RELEVANCE OF THE USE OF METHOD OF UNDETERMINE COEFFICIENTS FOR SOLVING DIFFERENTIAL EQUATIONS

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CHAPTER ONE

1.0 INTRODUCTION

1.1 BACKGROUND OF STUDY

The method of undetermined coefficient has been used as a tool for solving a particular non homogeneous differential equation.

One of the main advantages of this method is that it reduces the problem down to an algebra problem.  The algebra can get messy on occasion, but for most of the problems it will not be terribly difficult.  Another nice thing about this method is that the complementary solution will not be explicitly required, although as we will see knowledge of the complementary solution will be needed in some cases and so we’ll generally find that as well.

There are two disadvantages to this method.  First, it will only work for a fairly small class of g(t)’s.  The class of g(t)’s for which the method works, does include some of the more common functions, however, there are many functions out there for which undetermined coefficients simply won’t work.  Second, it is generally only useful for constant coefficient differential equations.

The method is quite simple.  All that we need to do is look at g(t) and make a guess as to the form of YP(t) leaving the coefficient(s) undetermined (and hence the name of the method).  Plug the guess into the differential equation and see if we can determine values of the coefficients.  If we can determine values for the coefficients then we guessed correctly, if we can’t find values for the coefficients then we guessed incorrectly.

It’s usually easier to see this method in action rather than to try and describe it, so let’s jump into some examples.

1.2 STATEMENT OF RESEARCH PROBLEM

What really instigated the study was due to over involvement of integration in solving differential equations by other method. There have been some problems in solving non-homogenous differential equation using other methods.

 

 

Although there are two disadvantages to this method; first, it will only work for a fairly small class of g(t)’s.  The class of g(t)’s for which the method works, does include some of the more common functions, however, there are many functions out there for which undetermined coefficients simply won’t work.   Second, it is generally only useful for constant coefficient differential equations.

The study will use the problems below to test for the efficiency of the use of the method of undetermined coefficient in solving differential equations:

1. 

2. 

3. 

1.3 AIMS AND OBJECTIVES OF STUDY

The main aim of the research work is to examine the relevance of the method of undetermined coefficient in solving differential equations. Other specific objectives of the study include:

1.  To examine the efficiency of the use of undetermined coefficient in solving differential equation

2.  To it reduces differential equation down to an algebra problem

3.  To determine the factors affecting the efficiency of the use of the method of undetermined coefficient in solving differential equation

4.  To proffer solution to the above problems

1.4 RESEARCH QUESTIONS

The study came up with research questions so as to ascertain the above stated objectives. The research questions for the study are stated below as follows:

1.  What is the efficiency of the use of undetermined coefficient in solving differential equation?

2.  How the method of undetermined coefficient does reduce differential equation down to an algebra problem?

3.  What are the factors affecting the efficiency of the use of the method of undetermined coefficient in solving differential equation?

1.5 SIGNIFICANCE OF STUDY

The study on the relevance of the method of undetermined coefficient will be of immense benefit to the entire mathematics department as it will solve differential equations using the method of undetermined coefficient. Finally the study will be of great importance to other research students that wishes to carryout similar research on the above topic.

1.6 SCOPE OF STUDY

The study will be limited to only the relevance of the use of the method of undetermined coefficient in solving second order differential equation

1.7 DEFINITION OF TERMS

DIFFERENTIAL EQUATION: an equation involving derivatives of a function or functions.

COEFFICIENT: a numerical or constant quantity placed before and multiplying the variable in an algebraic expression (e.g. 4 in 4x y)

RELEVANCE: the quality or state of being closely connected or appropriate.


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