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 , where 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 the radius and interval of convergence of a power series via the ratio test, and explain the significance of convergence.
- Perform algebraic (addition, subtraction, index shift) and calculus (esp. differentiation) operations on power series.
- Determine if a function is analytic at a point and determine if a point is an ordinary point of a differential equation.
- Solve linear differential equations via power series near ordinary points.
- Read Polking, Boggess and Arnold, Sections 11.1 and 11.2.
- Watch pencasts (linked above).
- Check yourself before you wreck yourself.
After class, please:
- Do these problems.