7 Linear second order differential equations

TABLE OF CONTENTS

1. Introduction

2. Existence theorems

3. Second order homogeneous linear differential equations

3.1 The Wronskian

4. Homogeneous equation with constant coefficients

5. The inhomogeneous linear equation

6. The method of undetermined coefficients

7. Reduction of order

8. Variation of parameters

LEARNING OBJECTIVES

1. The general linear differential equation of order n is described.
2. Some general theorems about the existence and nature of solutions of linear differential equations are proved.
3. The study is then specialised to second order linear differential equations. First, homogeneous equations are considered.
4. The Wronskian is introduced and condition for linear independence of solutions obtained.
5. Specialisation to homogeneous equation with constant coefficients is studied.
6. Next the inhomogeneous linear equations are considered.  The principle of superposition is enunciated.
7. The method of undetermined coefficients for solving inhomogeneous equations with constant coefficients is described.
8. Next the method of reduction of order for solving inhomogeneous equation, when a solution of the complimentary equation is known, is described.
9. Finally a very powerful method, method of variation of parameters, is described.

L i n e a r  S e c o n d  O r d e r  D i f f e r e n t i a l  E q u a t i o n s

1. Introduction

In this module we study a particularly important class of second order equations, the linear second order differential equations. Because of their many applications in science and engineering, linear second order differential equation have historically been the most thoroughly studied class of differential equations. Research on the theory of these equations continues to this day and new results keep on cropping up. We will first consider some general theorem which are based on the properties of linear systems and are applicable to linear differential equations of any order.

The most general linear differential equation can be written as

If the right hand side of the equation is zero, i.e., there is no term which is independent of y, the equation is said to be homogeneous otherwise it is called inhomogeneous. Using the idea of a differential operator, the equation may be put in the alternative form as

Here symbolically

2. Existence Theorems

The expression

is called linear differential operator of order n.  The differential equation

is the associated homogeneous equation corresponding to equation (2). It is also sometimes called the reduced equation.

The following theorems describe some of the important properties of the operator L and the inhomogeneous and homogeneous equations (2) and (5) respectively.

If n linearly independent solutions of the homogeneous equation (5) are known, then the solution

is the complete primitive of the homogeneous equation.  The n constants can be chosen by the requirement

The constants so obtained are unique since, according to the fundamental theorem, with these conditions the solution is unique.

Some examples from second order equations

3. Second order homogeneous linear differential equations

3.1 The Wronskian

Proof

Differentiate equation (14) with respect to x and use equation (13)

The function W defined by equation (14) is called the Wronskian and the relation (15) is called Abel’s formula.

The Wronskian is usually written in the alternative form of a determinant:

The Wronskian is zero at x = 0. Hence two linearly independent solutions will exist in the intervals (−∞, 0) and (0, ∞).

This verifies Abel’s formula.

4.  Homogeneous equation with constant coefficients

Substituting this proposed solution in the above equation, we have the characteristic equation

There are three cases to be considered:

Example

Hence the general solution is

5.  The inhomogeneous linear equation

We now consider the inhomogeneous equation

The following theorem which is the counterpart of the corresponding theorem for the homogeneous equation is about the uniqueness of the solution of an initial value problem:

Sometimes it is possible to find a particular solution of the given inhomogeneous equation by inspection. Then the complete solution to the problem can be obtained.

Examples

5.1 Principle of superposition

6.  The method of undetermined coefficients We now consider equations of the form

Example

7.  Reduction of order

This method attempts to find a solution of the general second order inhomogeneous equation

we try a solution for the inhomogeneous equation of the form

reminiscent of the method adopted for the first order equations.  Then

Plugging these into equation (29), we have

Examples

    

8. Variation of parameters

Another powerful method for finding solution of second order linear differential equation is the method of variation of parameters. For this we need the fundamental set of solutions of the complementary equation. This may seem to be unnecessary as the method of reduction of order needs only one solution of the complementary equation. So why use a method which requires both the solutions. It is usually much easier to apply than the method of reduction of order. Secondly this method can be generalized to higher order equations unlike the method of reduction of order.

Once again we consider equation (29)

Then

Then

Substituting these expressions for the first and second derivatives into equation (29) we have

Examples

Hence the general solution is

SUMMARY

• We begin this module with a description of the general linear differential equation of order n. We enunciate and prove some general theorems about the existence and nature of solutions of linear differential equations.
• After that we specialise to second order linear differential equations, our main topic in this module.
• We first deal with homogeneous equations.
• We introduce the Wronskian and obtain condition for linear independence of solutions.
• Next we specialise to homogeneous equation with constant coefficients and describe special method for such equations.
• We then come to inhomogeneous linear equations.
• The method of undetermined coefficients for solving inhomogeneous equations with constant coefficients is described.
• Next the method of reduction of order for solving inhomogeneous equation, when a solution of the complimentary equation is known, is described.
• Finally a very powerful method, that of variation of parameters, is described.