# Power series solutions

Get inspired:

Not all differential equations have solutions that can be expressed in terms of elementary functions that you learn in calculus. Some of the most intriguing examples are the Bessel functions, which are defined as the solutions to the differential equation $x^2 y'' + xy' + (x^2 - \alpha^2)y = 0$, where $\alpha$ is a parameter. These functions are most easily expressed as a power series. Bessel functions describe, among other things, the beautiful vibration patterns of circular drums.

By the end of this lesson, you should be able to:

Before class:

In class: