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CHAPTER 4

Series Solutions of Ordinary

Differential Equations

4.1 GENERAL SERIES SOLUTIONS
The purpose of this chapter is to present a method of obtaining solutions of linear second-order
ordinary differential equations in the form of Taylor series’. The methodology is then used to
obtain solutions of two special differential equations, Bessel’s equation and Legendre’s equation.
Properties of the solutions—Bessel functions and Legendre functions—which are extensively
used in solving problems in mathematical physics, are discussed briefly. Bessel functions are
used in solving both diffusion and vibrations problems in cylindrical coordinates. The functions
R(λ n r) in Example 3.4 at the end of Chapter 3 are called Bessel functions. Legendre functions
are useful in solving problems in spherical coordinates. Associated Legendre functions, also
useful in solving problems in spherical coordinates, are briefly discussed.

4.1.1 Definitions
In this chapter we will be concerned with linear second-order equations. A general case is

a(x)u + b(x)u + c (x)u = f (x) (4.1)

Division by a(x)gives

u + p(x)u + q(x)u = r(x) (4.2)

Recall that if r(x) is zero the equation is homogeneous. The solution can be written as the sum of
a homogeneous solution u h (x)and a particular solution u p (x). If r(x)iszero, u p = 0. The nature
of the solution and the solution method depend on the nature of the coefficients p(x)and q(x).
If each of these functions can be expanded in a Taylor series about a point x 0 the point is said
to be an ordinary point and the function is analytic at that point. If either of the coefficients is
not analytic at x 0 ,the pointisa singular point.If x 0 is a singular point and if (x − x 0 )p(x)and
2
(x − x 0 ) q(x) are analytic, then the singularities are said to be removable and the singular point
is a regular singular point.Ifthisisnot thecasethe singular pointis irregular.