Mathematical Methods for Physics and Engineering - Matematica.NET

16.1 SECOND-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONScan be written as the sum of the solution to the homogeneous equation y c (x)(the complementary function) and any function y p (x) (the particular integral) thatsatisfies (16.5) and is linearly independent of y c (x). We have thereforey(x) =c 1 y 1 (x)+c 2 y 2 (x)+y p (x). (16.6)General methods for obtaining y p , that are applicable to equations with variablecoefficients, such as the variation of parameters or Green’s functions, were discussedin the previous chapter. An alternative description of the Green’s functionmethod for solving inhomogeneous equations is given in the next chapter. For thepresent, however, we will restrict our attention to the solutions of homogeneousODEs in the form of convergent series.16.1.1 Ordinary and singular points of an ODESo far we have implicitly assumed that y(x) isareal function of a real variablex. However, this is not always the case, and in the remainder of this chapter webroaden our discussion by generalising to a complex function y(z) ofacomplexvariable z.Let us therefore consider the second-order linear homogeneous ODEy ′′ + p(z)y ′ + q(z) =0, (16.7)where now y ′ = dy/dz; this is a straightforward generalisation of (16.1). A fulldiscussion of complex functions and differentiation with respect to a complexvariable z is given in chapter 24, but for the purposes of the present chapter weneed not concern ourselves with many of the subtleties that exist. In particular,we may treat differentiation with respect to z in a way analogous to ordinarydifferentiation with respect to a real variable x.In (16.7), if, at some point z = z 0 , the functions p(z) andq(z) are finite and canbe expressed as complex power series (see section 4.5), i.e.∞∑∞∑p(z) = p n (z − z 0 ) n , q(z) = q n (z − z 0 ) n ,n=0then p(z) andq(z) are said to be analytic at z = z 0 , and this point is called anordinary point of the ODE. If, however, p(z) orq(z), or both, diverge at z = z 0then it is called a singular point of the ODE.Even if an ODE is singular at a given point z = z 0 , it may still possess anon-singular (finite) solution at that point. In fact the necessary and sufficientcondition § for such a solution to exist is that (z − z 0 )p(z) and(z − z 0 ) 2 q(z) areboth analytic at z = z 0 . Singular points that have this property are called regularn=0§ See, for example, H. Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics, 3rdedn(Cambridge:Cambridge University Press, 1966), p. 479.533