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Chapter 6 : Series Solutions to Differential Equations

In this chapter we will finally be looking at nonconstant coefficient differential equations. While we won’t cover all possibilities in this chapter we will be looking at two of the more common methods for dealing with this kind of differential equation.

The first method that we’ll be taking a look at, series solutions, will actually find a series representation for the solution instead of the solution itself. You first saw something like this when you looked at Taylor series in your Calculus class. As we will see however, these won’t work for every differential equation.

The second method that we’ll look at will only work for a special class of differential equations. This special case will cover some of the cases in which series solutions can’t be used.

Here is a brief listing of the topics in this chapter.

Review : Power Series – In this section we give a brief review of some of the basics of power series. Included are discussions of using the Ratio Test to determine if a power series will converge, adding/subtracting power series, differentiating power series and index shifts for power series.

Review : Taylor Series – In this section we give a quick reminder on how to construct the Taylor series for a function. Included are derivations for the Taylor series of \({\bf e}^{x}\) and \(\cos(x)\) about \(x = 0\) as well as showing how to write down the Taylor series for a polynomial.

Series Solutions – In this section we define ordinary and singular points for a differential equation. We also show who to construct a series solution for a differential equation about an ordinary point. The method illustrated in this section is useful in solving, or at least getting an approximation of the solution, differential equations with coefficients that are not constant.

Euler Equations – In this section we will discuss how to solve Euler’s differential equation, \(ax^{2}y'' + b x y' +c y = 0\). Note that while this does not involve a series solution it is included in the series solution chapter because it illustrates how to get a solution to at least one type of differential equation at a singular point.