Power Series Solutions to Differential Equations

When do we use a power series approach to solve diff eq

a(x)y” + b(x)y’ + c(x)y = 0

• when a(x), b(x), and c(x) aren’t constants

What is the power series for y

What is the power series for y’

What is the power series for y”

What are the general steps for solving a power series diff eq

1. analyze the problem —> if they have y”, y’, or y

2. write out the general power series equation (in terms of x^n)

   1. Note that in most cases will need to account for the constant or 'x’ in front of the y

3. Using general equations combine them into 1 singular equation —> set them equal to 0

   1. then solve for Cn (recursive case)

   2. Write out the pattern of Cn

4. Find general case

Info

Process doesn’t always give a recursive formula

• can output a polynomial