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Unformatted text preview: 31 Power Series Solutions I: Basic Computational Methods When a solution to a differential equation is analytic at a point, then that solution can be rep- resented 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 use 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 33 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. 31.1 Basics General Power Series Solutions If it exists, a power series solution for a differential equation is just a power series formula y ( x ) = summationdisplay k = a k ( x x ) k for a solution y to the given differential equation in some open interval containing x . The series is a general power series solution if it describes all possible solutions in that interval. As noted in the last chapter (corollary 30.10 on page 616), if y ( x ) is given by the above power series, then a = y ( x ) and a 1 = y ( x ) . Because a general solution to a first-order differential equation normally has one arbitrary con- stant, we should expect a general power series solution to a first-order differential equation to also have a single arbitrary constant. And since that arbitrary constant can be determined by a given initial value y ( x ) , it makes sense to use a as that arbitrary constant. 8/14/2011 625 626 Power Series Solutions I: Basic Computational Methods On the other hand, a general solution to a second-order differential equation usually has two arbitrary constants, and they are normally determined by initial values y ( x ) and y ( x ) . Consequently, we should expect the first two coefficients, a and a 1 , to assume the roles of the arbitrary constants in our general power series solutions for second-order differential equations. The Two Methods, Briefly The basic ideas of both the algebraic method and the Taylor series are fairly simple. The Algebraic Method The algebraic method starts by assuming the solution y can be written as a power series y ( x ) = summationdisplay k = a k ( x x ) k with the a k s being constants to be determined. This formula for y is then plugged into the differential equation. By using a lot of algebra and only a little calculus, we then simplify the resulting equation until it looks something like summationdisplay n = bracketleftbig n th formula of the...
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