Mathematical Methods for Physics and Engineering : A Comprehensive Guide

16.1 SECOND-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS

can be written as the sum of the solution to the homogeneous equationyc(x)

(the complementary function) andanyfunctionyp(x) (the particular integral) that

satisfies (16.5) and is linearly independent ofyc(x). We have therefore

y(x)=c 1 y 1 (x)+c 2 y 2 (x)+yp(x). (16.6)

General methods for obtainingyp, that are applicable to equations with variable

coefficients, such as the variation of parameters or Green’s functions, were dis-

cussed in the previous chapter. An alternative description of the Green’s function

method for solving inhomogeneous equations is given in the next chapter. For the

present, however, we will restrict our attention to the solutions of homogeneous

ODEs in the form of convergent series.

16.1.1 Ordinary and singular points of an ODE

So far we have implicitly assumed thaty(x)isarealfunction of arealvariable

x. However, this is not always the case, and in the remainder of this chapter we

broaden our discussion by generalising to acomplexfunctiony(z)ofacomplex

variablez.

Let us therefore consider the second-order linear homogeneous ODE

y′′+p(z)y′+q(z)=0, (16.7)

where nowy′=dy/dz; this is a straightforward generalisation of (16.1). A full

discussion of complex functions and differentiation with respect to a complex

variablezis given in chapter 24, but for the purposes of the present chapter we

need not concern ourselves with many of the subtleties that exist. In particular,

we may treat differentiation with respect tozin a way analogous to ordinary

differentiation with respect to a real variablex.

In (16.7), if, at some pointz=z 0 , the functionsp(z)andq(z) are finite and can

be expressed as complex power series (see section 4.5), i.e.

p(z)=

∑∞

n=0

pn(z−z 0 )n,q(z)=

∑∞

n=0

qn(z−z 0 )n,

thenp(z)andq(z) are said to beanalyticatz=z 0 , and this point is called an

ordinary pointof the ODE. If, however,p(z)orq(z), or both, diverge atz=z 0

then it is called asingular pointof the ODE.

Even if an ODE is singular at a given pointz=z 0 , it may still possess a

non-singular (finite) solution at that point. In fact the necessary and sufficient

condition§for such a solution to exist is that (z−z 0 )p(z)and(z−z 0 )^2 q(z)are

both analytic atz=z 0. Singular points that have this property are calledregular

§See, for example, H. Jeffreys and B. S. Jeffreys,Methods of Mathematical Physics,3rdedn(Cam-

bridge: Cambridge University Press, 1966), p. 479.