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:
- Compute and explain the significance of radius and interval of convergence – [movie] [notes]
- Perform algebraic operations on power series – [movie] [notes]
- Determine if a point is an ordinary point of a differential equation – [movie] [notes]
- Solve linear differential equations via power series near ordinary points – [movie not available — read notes and textbook] [notes]
- Read Polking, Boggess and Arnold, Sections 11.1 and 11.2.
- Watch pencasts (linked above).
After class, please:
- Do post-class problems.