Mathematical Methods for Physics and Engineering - Matematica.NET

15.1 LINEAR EQUATIONS WITH CONSTANT COEFFICIENTSHigher-order recurrence relationsIt will be apparent that linear recurrence relations of order N>2 do not presentany additional difficulty in principle, though two obvious practical difficulties are(i) that the characteristic equation is of order N and in general will not have rootsthat can be written in closed form and (ii) that a correspondingly large numberof given values is required to determine the N otherwise arbitrary constants inthe solution. The algebraic labour needed to solve the set of simultaneous linearequations that determines them increases rapidly with N. We do not give specificexamples here, but some are included in the exercises at the end of the chapter.15.1.5 Laplace transform methodHaving briefly discussed recurrence relations, we now return to the main topicof this chapter, i.e. methods for obtaining solutions to higher-order ODEs. Onesuch method is that of Laplace transforms, which is very useful for solvinglinear ODEs with constant coefficients. Taking the Laplace transform of such anequation transforms it into a purely algebraic equation in terms of the Laplacetransform of the required solution. Once the algebraic equation has been solvedfor this Laplace transform, the general solution to the original ODE can beobtained by performing an inverse Laplace transform. One advantage of thismethod is that, for given boundary conditions, it provides the solution in justone step, instead of having to find the complementary function and particularintegral separately.In order to apply the method we need only two results from Laplace transformtheory (see section 13.2). First, the Laplace transform of a function f(x) is definedby¯f(s) ≡∫ ∞0e −sx f(x) dx, (15.31)from which we can derive the second useful relation. This concerns the Laplacetransform of the nth derivative of f(x):f (n) (s) =s n¯f(s) − s n−1 f(0) − s n−2 f ′ (0) − ···− sf (n−2) (0) − f (n−1) (0),(15.32)where the primes and superscripts in parentheses denote differentiation withrespect to x. Using these relations, along with table 13.1, on p. 455, which givesLaplace transforms of standard functions, we are in a position to solve a linearODE with constant coefficients by this method.501