At this point we have systematic techniques for solving the initial value problem a_{2}(x)y''+a_{1}(x)y'+a_{0}(x)y=0, y(x_{0})=y_{0}, y'(x_{0})=y_{0}^{*} when the second order differential equation has the form of a CauchyEuler equation or constant coefficients. In this chapter we develop techniques to find solutions of a_{2}(x)y''+a_{1}(x)y'+a_{0}(x)y=0, y(x_{0})=y_{0}, y'(x_{0})=y_{0}^{*} when a_{2}(x), a_{1}(x), and a_{0}(x) are arbitrary functions of x. We do this by using power series and base it on the techniques introduced in Chapters 1 and 3.
We state theorems that show when solutions exist in the form of Taylor series
If any of a_{2}(x), a_{1}(x), and a_{0}(x) are not continuous, or if a_{2}(x)=0 at some point, a Taylor series solution may not exist. However, with appropriate restrictions on the behavior of a_{1}(x)/a_{2}(x) and a_{0}(x)/a_{2}(x) at x_{0} we can modify our techniques to find general solutions of the form A knowledge of the first few terms in a series solution is useful in determining the behavior of the solution near a given initial value. Occasionally the infinite series solution is the series expansion of a familiar function. When this occurs we can write such solutions in simpler, and more common, terms. The functions with which we are familiar are usually defined by a simple formula. The techniques presented in this chapter are important because they allow us to introduce and define new functions—which occur naturally in many fields—as infinite series. This gives us a significant way to enlarge the family of familiar functions to include many important functions that cannot be defined by simple formulas. These functions are often designated Special Functions, and entire books are devoted to this subject.
