Mathematical Methods (10/24.539)

VII. Power Series Solution Method

Overview of the Method

Some Definitions

The Power Series method assumes homogeneous solutions of the form

where the am coefficients and the r exponent need to be determined as part of the solution procedure. As such, we need to work with infinite series of the form given in eqn. (7.1). Thus, we begin our discussion of the method with a few definitions establishing the proper terminology:

Analytic Function -- A function is said to be analytic at x = xo if it can be represented by a power series of (x - xo) with a radius of convergence R > 0.

Radius of Convergence -- R is the radius of convergence if a series converges for x in the range defined by and diverges for x in the range .

If the function f(x) is analytic at x = xo, then it can be written as

where the radius of convergence is given by

As an example, consider the following expansion for around the point xo = 0,

Therefore,

and the radius of convergence is given by

Thus this series converges for all values of x.

As another example, consider the following expansion,

If we let , then

Therefore,

Thus this particular series converges for .

Theorem #1 -- Power Series Solution

If the functions p(x), q(x), and f(x) in the differential equation

are analytic at x = xo, then every solution, y(x), is analytic at x = xo and it can be represented by a power series in powers of x - xo with radius of convergence R > 0. Therefore, we have a power series solution of the form given in the first part of eqn. (7.1), or

Theorem #2 -- Extended Power Series Solution Method

The differential equation,

with b(x), c(x), and f(x) analytic around x0 has at least one solution of the form given in the second part of eqn. (7.1), or

This is referred to as an extended power series with r chosen such that . A second independent solution may contain a logarithmic term if the roots are repeated or if they differ by an integer.

Relations for y' and y''

A function written in the form of a power series may be differentiated term by term. Therefore, based on the more general representation in eqn. (7.7), one has

and

and this list can be easily extended for higher-order derivatives.

Basic Solution Outline

Although somewhat tedious for most practical applications, a systematic procedure for the Power Series method can be identified, as follows:

1. Expand all the terms in the original differential equation in a power series about the point x = x0 (in practice, x0 = 0 in most cases).

2. Assume a solution of the form .

3. Substitute the assumed solution and its derivatives into the differential equation from Step 1 and shift all indices to that for the highest power of x with all the sums beginning with m = 0 (this is a very important step).

4. Collect terms with like powers and equate the coefficients to the right hand side coefficients (these will be zero for a homogeneous equation).

5. Evaluate r in the original expression such that (this gives the indicial equation).

6. Obtain a recurrence relation for general term, am.

7. If repeated roots are obtained for r or y1(x) and y2(x) are linearly dependent, then one should use the variation of parameters (reduction of order ) method to find a second linearly independent solution. In this case, simply let . Note that, although conceptually straightforward, this can often be quite tedious if y1(x) cannot be written in a simple closed form solution.

This procedure is illustrated in the remainder of this section for a series of three cases. These illustrate the range of typical situations that arise in practical applications. The following section on Special Functions also gives some further examples of the basic Power Series solution methodology.

10/24.539 Lecture Notes by Dr. J. R. White, UMass-Lowell (updated October 1998).

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