15.1 LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS
Higher-order recurrence relationsIt 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 methodHaving 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 transformtheory (see section 13.2). First, the Laplace transform of a functionf(x) is defined
by
̄f(s)≡∫∞0e−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.