Solving differential equations using power series

 Objective: Solve differential equations using power series.

Let's consider the differential equation y'(x) = y. Recall that this is a first order separable equation and its solution is y = Ce^x. For most differential equations there are not any analytical tools to solve them. Power series are an extremely useful tool to solve them. The technique we use is to look for a solution of the form 

Then we determine the coefficients in order to solve the differential equation.

Example

Solution

Using the uniqueness of power series representations, we know that these series can only be equal if their coefficients are equal. Therefore:

Practice