Chapter 2 - Power Series Solution

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2.1 Review of Power and Taylor Series

Taylor Series are an example of a power series and is a generalisation of MacLaurin series.

In order to talk about power series solutions to ODEs we need to series to converge. We can check for convergence using the ratio test.

2.2 Power Series Solutions Techniques

Consider the homogeneous second order linear ODE

a(x)y'' + b(x)y' + c(x)y = 0 (2.1)

where a(x), b(x) and c(x) are polynomials.

We look for power series solutions to the ODE near point x_0. The power series solution is valid if the series converges.

2.2.1 Ordinary Points

In this section we assume a(x) does not equal 0 in equation 2.1

Definition 2.2 - A point x_0 in R is called an ordinary point of equation 2.1 if a(x_0) does not equal zero and otherwise x_0 is called a singular point.

We can also use the recurrence relation of the power series to determine even and odd powers of x.

Definition 2.5 - A point x_0 in R is a regular singular point of equation 2.1 if the limits of b(x)/a(x) and c(x)/a(x) (check notes for exact formula) are both finite. If these conditions do not hold then the singular point is irregular.

Euler's Equation - x^2y'' + xay' + by = 0 where a, b are in R. This is simple to solve using y(x) = x^r.