Prelude to Series Solutions of Linear Second Order Equations

In this Chapter, we study a class of second order differential equations that occur in many applications, but cannot be solved in closed form in terms of elementary functions. Here are some examples:

1. Bessel’s equation which occurs in problems displaying cylindrical symmetry, such as diffraction of light through a circular aperture, propagation of electromagnetic radiation through a coaxial cable, and vibrations of a circular drum head. $$x^2y''+xy'+(x^2-\nu^2)y=0, \nonumber$$
2. Airy’s equation which occurs in astronomy and quantum physics. $$y''-xy=0, \nonumber$$
3. Legendre’s equation which occurs in problems displaying spherical symmetry, particularly in electromagnetism. $$(1-x^2)y''-2xy'+\alpha(\alpha+1)y=0, \nonumber$$

These equations and others considered in this chapter can be written in the form

$$P_0(x)y''+P_1(x)y'+P_2(x)y=0, \label{A}$$

where $P_0$, $P_1$, and $P_2$ are polynomials with no common factor. For most equations that occur in applications, these polynomials are of degree two or less. We’ll impose this restriction, although the methods that we’ll develop can be extended to the case where the coefficient functions are polynomials of arbitrary degree, or even power series that converge in some circle around the origin in the complex plane.

Since Equation $\ref{A}$ does not in general have closed form solutions, we seek series representations for solutions. We’ll see that if $P_0(0)\ne0$ then solutions of (A) can be written as power series $$y=\sum_{n=0}^\infty a_nx^n \nonumber$$ that converge in an open interval centered at $x=0$.