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5.1 Power Series Method

The power series method is the standard method for solving linear ODEs with variable coefficients. It gives solutions in the form of power series. These series can be used for computing values, graphing curves, proving formulas, and exploring properties of solutions, as we shall see. In this section we begin by explaining the idea of the power series method.

From calculus we remember that a power series (in powers of xx0) is an infinite series of the form

image

Here, x is a variable. a0, a1, a2, … are constants, called the coefficients of the series. x0 is a constant, called the center of the series. In particular, if x0 = 0, we obtain a power series in powers of x

image

We shall assume that all variables and constants are real.


  

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