8 Series solution of differential equations

TABLE OF CONTENTS

1. Introduction

2. Solution near an ordinary point

3. Solution near a regular singularity 3.1 The series solution

3.2 Convergence of the series solution near a regular singularity

4. Solution for large z

5. Solution when roots are equal or differ by integer

6. Equation with three regular singularities

LEARNING OBJECTIVES

1. Series method of solution, the most general for linear equations, is introduced.

2. First, method for series solution near an ordinary point of a differential equation is described.

3. Next the case of a regular singularity is taken up. First the case of real distinct roots of the indicial equation is described and the convergence of the series solution established.

4. Then the case of solution valid for large z is considered.

5. Next the case in which the roots of the indicial equation are equal or differ by an integer is taken up.

6. Finally solution of equations with three regular singular points is studied.

S e r i e s  s o l u t i o n  o f  d i f f e r e n t i a l  e q u a t i o n s

1. Introduction

There are many applications that lead to linear second order differential equations that cannot be solved by any of the analytical methods that we have discussed in the last module. Throughout physics and engineering we deal with problems related to potential theory, wave motion, heat transfer and diffusion. These are described in terms of partial differential equation of the elliptic, parabolic and hyperbolic type. When simplified by the method of separation of variables, they lead to second order linear differential equations which form the backbone of classical physics. Well known examples are the Bessel equation, the Legendre equation, the airy equation etc. The solutions of these equations cannot be expressed in terms of polynomials, rational functions, trigonometric or exponential functions or other elementary functions. The solution can be expressed in terms of infinite power series, and we now aim to study the power series method of finding solutions of these equations. Quite often we are interested in the solution of these classical equations in the complex domain. To emphasize this fact we will take z as our independent variable rather than x as we have been doing so far.

Before we embark on finding a series solution of second order linear equations, we first wish to see whether an analytic solution to the equation exists or not. We consider the equation

Examples

1.   For the Legendre equation

2.   For the Bessel equation

z = 0 is the only singular point.

3.      For the Airy’s equation

all points are ordinary.

2. Solution near an ordinary point

The radius of convergence of each series is at least equal to R. We now try to find a power series solution of the equation of the form [The index i is not to be confused with the complex number = √−1]

by substituting equations (2) and (3) into (1) and comparing the coefficients of various powers of z.  Thus we have

where

Similarly

where

Example-1

As an illustration we consider a simple equation where the result is well known:

Here all points are ordinary. The solution is known to be in terms of sine and cosine functions. Let us try a series solution of the form

On substituting in the equation above and comparing coefficients of various powers of z, we have

Example-2

It is not always possible to solve the recurrence relation to get the coefficients explicitly as functions of the index i. But we can calculate as many coefficients as we like. As an example consider

The point z = 0 is an ordinary point, so we try a solution of the form

Substituting in the equation above and comparing the coefficients of various power of z, we obtain

3. Solution near a regular singularity

Examples

1.  For the Legendre equation

2.   For the Bessel equation

The singular point at z = 1 is a regular singular point.

3.1 The series solution

Let us assume a solution of the form

If we substitute equations (9) and (10) into the differential equation (1) and compare the coefficients of various powers of z, we obtain the following recurrence relation:

where

The nature of the roots of the indicial equation is very important. If the indicial equation has two distinct roots, real or complex, that do not differ by an integer, we obtain two linearly independent solutions, one corresponding to each root. If the roots are equal or differ by an integer we obtain only one formal solution. Let us consider the first case for the time being and study the convergence of the series solutions.

3.2 Convergence of the series solution near a regular singularity

Like in the case of solution near an ordinary point, in this case also the solution is only formal till the convergence of the series is established. For the solution to be meaningful the series must either terminate or have a non-zero radius of convergence. If the series terminates there is nothing to prove; so let us assume that the series does not terminate. The proof of convergence is on lines similar that for the case of ordinary point. If a₁ is a root of the indicial equation (13), then from equation (12)

 where K is the larger of the two, M and N. If we now substitute these bounds, equation (15), into equation (14) we get

4. Solution for large z

So far we have talked of ordinary points or singular points for finite values of the variable z. It is also important to know about the nature of the solution at infinity. The nature of the solution will depend on whether the point at infinity is an ordinary point, a regular singular point or an irregular singular point. For this study we make the transformation u = 1/z and then study the behaviour of the resulting equation near u = 0. Making this transformation in equation (1) we obtain

The point at infinity is an ordinary point if

are regular at the origin, i.e., if

are regular at infinity. The complete solution of the equation in this case will be of the form

The point u = 0 is a regular singular point if

are not equal or do not differ by an integer, the two linearly independent solutions of the differential equation (1) in the neighbourhood of infinity are

5. Solution when roots are equal or differ by integer

The function u satisfies the equation

Since we are interested in the second linearly independent solution only we choose a = 0, b = 1. Hence the second solution to our differential equation is

On using equations (25) and (26) in (24) we get

6.  Equation with three regular singularities

Suppose the differential equation (1)

Example

The associated Legendre equation is

SUMMARY

• In this module we take up the series method of solving second order linear differential equations. We define ordinary, singular and regular singular points of a differential equation. In this module we allow complex solutions of a complex variable.
• We first describe the method near an ordinary point of a differential equation and establish the convergence of the series solution obtained.
• Next we take up the case of a regular singularity. We describe first the case of real distinct roots of the indicial equation and study the convergence of the series solution obtained.
• The point at infinity is of importance in such cases; the convergence of the series solution valid for large z is discussed.
• Next we take up the case in which the roots of the indicial equation are equal or differ by an integer. We see that in this case the solution has a logarithmic singularity in general.
• Lastly we briefly study the important case of equations with three regular singular points.