Series Solutions of Second Order Linear Differential Equations (Intro)

While I won't review any concepts about power series here in this page, there is one important topic that we need to go over before we begin to use the "Power Series Method" to find solutions to 2nd order differential equations. I will assume that you have prior knowledge of summation notation and the basic concept of what a Power Series is. Before we continue however, let's discuss the concept of shifting the index of summation of an infinite series.

Shifting the index of summation:

In certain cases, it is desirable to shift the index of summation or rewrite the generic term of a series expression. Let's take a look at a particular example

Rewrite the following expression as a sum whose generic term is x^k

$$\sum_{n=0}^{\infty} a_n x^{n+2}$$

We start out with a substitution where k is the value of the exponent in the x term:

$$k = n+2 $$

...and correspondingly:

$$n = k-2 $$

keep in mind that k is just a dummy variable (you could use any letter/symbol). What IS important is that this substitution will allow us to shift the index of summation and correspondingly rewrite the generic term (the x variable) in terms of a different power. We will see why this is important when we go on to solve 2nd order differential equations using series solution methods.

We now proceed to rewrite all "n" values in the original expression in terms of "k" using the above two substitutions. Where the original expression started at the index n=0, it will now start at the index of k=2 (since k=n+2, n being zero). Additionally, the n subscript for the variable "a" was "n" and is now written in terms of "k" (n=k-2). All of this brings us to the following expression:

$$\sum_{k=2}^{\infty} a_{k-2} x^{k} $$

Continue on to Power Series example