## ABSTRACT

When a solution to a differential equation is analytic at a point, then that solution can be represented by a power series about that point. In this and the next chapter, we will discuss when this can be expected, and how we might employ this fact to obtain usable power series formulas for the solutions to various differential equations. In this chapter, we will concentrate on two basic methods — an “algebraic method” and a “Taylor series method” — for computing our power series. Our main interest will be in the algebraic method. It is more commonly used and is the method we will extend in Chapter 35 to obtain “modified” power series solutions when we do not quite have the desired analyticity. But the algebraic method is not well suited for solving all types of differential equations, especially when the differential equations in question are not linear. For that reason (and others) we will also introduce the Taylor series method near the end of this chapter.