So far every solution has been written as a combination of functions already familiar from calculus. That short list runs out quickly once the coefficients vary with . This chapter replaces the search for a closed-form answer with a search for a power series, which turns the differential equation into a recurrence relation among the coefficients.
Why Series Are Necessary
So far, we have been able to express the solution of a differential equation as a combination of functions we know from calculus: , , , the trigonometric functions, the inverse trigonometric functions, and so on. It is not a very long list.
Such a list may be enough for linear differential equations with constant coefficients, but as soon as we explore linear equations with variable coefficients we find it far too restrictive. In fact, many important linear differential equations arising throughout science and engineering, setting aside those with constant coefficients, have no solutions expressible in terms of elementary functions. Among them are Legendre's equation, Bessel's equation, and the hypergeometric equation.
In this chapter we introduce two methods for solving linear differential equations with variable coefficients.
In the first method, we write the solution as a power series. A power series in is an infinite series of the form
where is a variable and and are constants. The second method generalizes the first: we multiply a power series by a logarithmic function or by a fractional power of .
The series solutions that have extensive applications are frequently used to define new functions. The properties of these functions are then investigated by means of their series. All such functions, along with various others not appearing in calculus, are known as special functions. The Bessel functions, Legendre polynomials, Airy functions, and hypergeometric functions all arise this way, and they are no less legitimate than or , which are themselves defined by power series.
What This Chapter Covers
| Section | What you will learn |
|---|---|
| 6.1 Power Series | Convergence of series, the radius of convergence, and the two operations that make the method work: term-by-term differentiation and matching coefficients. |
| 6.2 Power Series Solutions | Ordinary points, the existence theorem, and the mechanics of substituting a series into an equation to obtain a recurrence relation. |
| 6.3 The Method of Frobenius | Regular singular points, the indicial equation, and the three cases determined by its roots. |
The Idea in One Paragraph
Assume the unknown solution is with the coefficients unknown. Differentiate term by term to get series for y' and y'', substitute all three into the equation, and collect like powers of . Because a power series is identically zero only when all its coefficients vanish, each power of gives one equation. Those equations link to earlier coefficients, forming a recurrence relation that generates every coefficient from and . Those two remain free, and they are exactly the two arbitrary constants a second-order equation must have.
Frequently Asked Questions
Is a series solution a real solution?
Yes. Within its radius of convergence a power series defines a genuine, infinitely differentiable function that satisfies the equation exactly. There is nothing approximate about it. What is approximate is any truncation of the series to finitely many terms, which is a separate matter.
Why can't I just use elementary functions?
Because most linear equations with variable coefficients have no elementary solutions, and this can be proved rather than merely suspected. Airy's equation y'' = xy is about as simple as such an equation gets, and its solutions are not expressible in elementary terms. Series are not a fallback; for these equations they are the answer.
What is the difference between the two methods?
The power series method of Section 6.2 works at an ordinary point, where the coefficients of the equation are well behaved. The Frobenius method of Section 6.3 extends it to a regular singular point, where the coefficients blow up in a controlled way, by allowing an extra factor with possibly fractional or negative, and sometimes a logarithm.
How do I know how many terms to compute?
For a formula, compute enough terms to recognize the pattern and write down a closed form for . For a numerical answer near , compute until the terms become smaller than your tolerance, which happens quickly close to the center and slowly near the edge of the interval of convergence.
