Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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15.1 LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS


Higher-order recurrence relations

It will be apparent that linear recurrence relations of orderN>2 do not present


any additional difficulty in principle, though two obvious practical difficulties are


(i) that the characteristic equation is of orderNand in general will not have roots


that can be written in closed form and (ii) that a correspondingly large number


of given values is required to determine theNotherwise arbitrary constants in


the solution. The algebraic labour needed to solve the set of simultaneous linear


equations that determines them increases rapidly withN. We do not give specific


examples here, but some are included in the exercises at the end of the chapter.


15.1.5 Laplace transform method

Having briefly discussed recurrence relations, we now return to the main topic


of this chapter, i.e. methods for obtaining solutions to higher-order ODEs. One


such method is that of Laplace transforms, which is very useful for solving


linear ODEs with constant coefficients. Taking the Laplace transform of such an


equation transforms it into a purelyalgebraicequation in terms of the Laplace


transform of the required solution. Once the algebraic equation has been solved


for this Laplace transform, the general solution to the original ODE can be


obtained by performing an inverse Laplace transform. One advantage of this


method is that, for given boundary conditions, it provides the solution in just


one step, instead of having to find the complementary function and particular


integral separately.


In order to apply the method we need only two results from Laplace transform

theory (see section 13.2). First, the Laplace transform of a functionf(x) is defined


by


̄f(s)≡

∫∞

0

e−sxf(x)dx, (15.31)

from which we can derive the second useful relation. This concerns the Laplace


transform of thenth derivative off(x):


f(n)(s)=sn ̄f(s)−sn−^1 f(0)−sn−^2 f′(0)−···−sf(n−2)(0)−f(n−1)(0),
(15.32)

where the primes and superscripts in parentheses denote differentiation with


respect tox. Using these relations, along with table 13.1, on p. 455, which gives


Laplace transforms of standard functions, we are in a position to solve a linear


ODE with constant coefficients by this method.

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