IN THIS CHAPTER we study a class of second order differential equations that occur in many applications, but can’t 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.
(2) Airy’s equation,
which occurs in astronomy and quantum physics.
(3) Legendre’s equation
which occurs in problems displaying spherical symmetry, particularly in electromagnetism.
These equations and others considered in this chapter can be written in the form
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 (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
that converge in an open interval centered at \(x=0\).
SECTION 7.1 reviews the properties of power series.
SECTIONS 7.2 AND 7.3 are devoted to finding power series solutions of (A) in the case where \(P_0(0)~\ne~0\). The situation is more complicated if \(P_0(0)=0\); however, if \(P_1\) and \(P_2\) satisfy assumptions that apply to most equations of interest, then we’re able to use a modified series method to obtain solutions of (A).
SECTION 7.4 introduces the appropriate assumptions on \(P_1\) and \(P_2\) in the case where \(P_0(0)=0\), and deals with Euler’s equation
where \(a\), \(b\), and \(c\) are constants. This is the simplest equation that satisfies these assumptions.
SECTIONS 7.5 –7.7 deal with three distinct cases satisfying the assumptions introduced in Section 7.4. In all three cases, (A) has at least one solution of the form
where \(r\) need not be an integer. The problem is that there are three possibilities – each requiring a different approach – for the form of a second solution \(y_2\) such that \(\{y_1,y_2\}\) is a fundamental pair of solutions of (A).