Throughout these pages I will assume that you are familiar with power series and the concept of the radius of convergence of a power series.

Let us consider the example of the second order differential equation

You probably already know that the solution is given by ; recall that this function has as its Taylor series

cos(2*t*) = 1 - + - +...,

and that this representation is valid for all .

Do you remember how to compute the Taylor series expansion for a given function?

If everything works out, then

where

Here

What does this mean? To compute the Taylor series for the solution to our differential equation, we just have to compute its derivatives. Note that the initial conditions give us a head start: *y*(0)=1, so *a*_{0}=1. *y*'(0)=0, so *a*_{1}=0.

We can rewrite the differential equation as

so in particular

Consequently

Next we differentiate

on both sides with respect to

so in particular

yielding

We can continue in this fashion as long as we like:
Differentiating

yields

in particular

so

We expect that this Taylor polynomial is reasonably close to the solution *y*(*t*) of the differential equation, at least close to *t*=0. In the picture below, the solution
is drawn in red, while the power series approximation
is depicted in blue:

The method outlined works also in theory for non-linear differential equations, even though the computational effort usually becomes prohibitive after the first few steps. Let's consider the example

We try to find the Taylor polynomials for the solution, of the form

where

The initial conditions yield

and plug in

consequently

Next we differentiate

on both sides

so in particular

yielding

Let's continue a little bit longer: We differentiate

to obtain

Consequently

Differentiating

yields

You can check that

The fifth degree Taylor polynomial approximation to the solution of our differential equation has the form

We again expect that this Taylor polynomial is reasonably close to the solution *y*(*t*) of the differential equation, at least close to *t*=0. In the picture below, the solution, as computed by a numerical method, is drawn in red, while the power series approximation
is depicted in blue:

The next sections will develop an organized method to find power series solutions for second order linear differential equations. Here are a couple of examples to practice what you have learned so far:

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