Differential Equations - Power Series

In order to solve a differential equation by the use of power series, we simply find the coefficients then attempt to match the created series to a maclaurin or taylor expansion of a known function. Let us try solving for the solution of the following differential equation.

y'' - xy' + y = 0

We substitue in for the differentials with a power series that represent the differential.

Which will give us the following equation

Now we set y to 0 to solve around the origin, which is an ordinary point in this case, as well as changing the indices from m to k. For the first term, we set 

m - 2 = k

For the second and third terms we just set

m = k

Giving us the equation

Now we need to match our indices so we'll write the first term where k = 0, then write the combined sum starting from k = 1.

Which we can then solve for the recursion relation.

Now that we have the recursion relation we can then solve for the first few coefficient terms in the solution.

C₂ = -1/2 C₀
C₃ = 0
C₄ = 1/12 C_{2 =} -1/24 C₀

Which gives us the following as the solution to this power series problem