Mathematics for Physicists

15 Series solutions of ordinary differential equations

In this chapter we will extend our discussion of ordinary differential equations (ODEs) to include linear second-order equations of the form

(15.1)

where the coefficients p(x) and q(x) are no longer restricted to constants, but may be arbitrary functions. Many ways of solving such equations apply only to a very limited range of equations, or require some prior knowledge of the solution. One such method will be mentioned at the end of Section 15.1.3. Otherwise we will confine ourselves to the most important method, which is to seek a solution in the form of a power series expansion about a particular point x = x₀.

This method is introduced in Section 15.1 and then, after a brief discussion of differential operators and eigenvalue equations, illustrated by applying it to two eigenvalue equations that are particularly important in physics.

15.1 Series solutions

The existence of solutions in the form of power series expansions about a particular point x = x₀ depends on the behaviours of p(x) and q(x) in the neighbourhood of x₀. Three types of behaviour need to be distinguished. If p(x) and q(x) are finite, single-valued and differentiable, then x₀ is called a regular or ordinary point and (15.1) is said to be regular at x = x₀. In this case, the limits of p(x) and q(x) as x → x₀ both exist, that is, are finite. If either ...