Methods of Solution to Second Order Linear Differential Equation With Variable Coefficients

Methods of Solution to Second Order Linear Differential Equation With Variable Coefficients

CHAPTER ONE

1.0 INTRODUCTION

1.1 BACKGROUND OF THE STUDY

The general second order homogeneous linear differential equation with constant coefficients is

Ay’’ + By’ + Cy = 0,

where y is an unknown function of the variable x, and A, B, and C are constants. If A = 0 this becomes a first order linear equation, which in this case is separable, and so we already know how to solve. So we will consider the case A 0. If we want, we can the divide through by A and obtain the equivalent equation

y″ + by′ + cy = 0,

where b = B/A and c = C/A (that is if we have nothing better to do, and like the first coefficient to be equal to 1)

Linear with constant coefficients means that each term in the Left Hand Side of the equation is a constant times y or a derivative of y. Homogeneous means that we exclude equations like

Ay″ + By′ + Cy = f(x)

which can be solved, in certain important cases, by an extension of the methods we will study here. Here we only will solve the case where the Right Hand Side f(x) is identically 0. Homogeneous also means that the constant function y = 0 is always a solution to the equation.

By now we know to expect 2 degrees of freedom in the solution of this second order equation, i.e., the general family of solutions should have two arbitrary constants. We call the general family of solutions for short the general solution. That means that to find our general solution we have to find two independent functions y = f1(x) and y = f2(x) which are solutions, and then the general solution will be

y = C1 · f1(x) + C2 · f2(x). Now If y = f1(x)

and y = f2(x) are indeed solutions, one can check by plugging in that y = C1 · f1(x) + C2 · f2(x) will be a solution. The fact that all solutions are of this form (i.e., we haven’t missed any solution) is harder to ascertain, but nevertheless true. Notice that of course we do need f1(x) and f2(x) to be independent. What does it mean for two functions to be independent? It means that one of them is not equal to a constant times the other. For example the functions f1(x) = e x

and f2(x) = 2e x are not independent, because f2(x) = 2 · f1(x). On the other hand, for example the functions g1(x) = e x and g2(x) = e 2x are independent (although we do not prove it, it seems intuitively clear that they are independent).

1.2 STATEMENT OF THE PROBLEM

The study will try to solve the problems below:

y″ -xy′ + 2y = 0

y″ + y′ = 0 and x2y″ + y′ + xy = 0 using power series and Frobenius method

1.3 AIM AND OBJECTIVES OF THE STUDY

The main aim of the research work is determine the methods of solution to second order linear differential equation with variable coefficients. Other specific objectives of the study are:

1. to determine the solution around the origin for homogenous and non-homogenous second order differential equation with variable coefficients
2. to determine the solution at other points
3. to investigate on factors affecting methods of solution to second order linear differential equation with variable coefficients
4. to determine the difference in efficiency of the methods of solution to second order linear differential equation with variable coefficients

1.4 SIGNIFICANCE OF THE STUDY

The study on the methods of solution to second order linear differential equation with variable coefficients will be of immense benefit to the mathematics department in the sense that the study will determine the solution around the origin for homogenous and non-homogenous second order differential equation with variable coefficients, the solution at other points and the difference in efficiency of the methods of solution to second order linear differential equation with variable coefficients. The study will serve as a repository of information to other researchers that desire to carry out similar research on the above topic. The study will contribute to the body of the existing literature on solution to second order linear differential equation with variable coefficients

1.5 SCOPE OF THE STUDY

The study on the methods of solution to second order linear differential equation with variable coefficients will focus on two methods (power series solution to DE and method of Frobenius.

1.6 DEFINITION OF TERMS

ODE: differential equation

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