1550078481-Ordinary_Differential_Equations__Roberts_

236 Ordinary Differential Equations

Let f(x) and g(x) be continuous functions on [O, oo) and assume they have

Laplace transforms F(s) = .C[f(x)] and G(s) = £[g(x)]. Since the Laplace

transform is a linear operator, for arbitrary constants c 1 and c2,

(9) £[ci.f(x) + c 2 g(x)] = c 1 £[f(x)] + c2£[g(x)] = c1F(s) + c2G(s).

Taking the inverse Laplace transform of (9), we find

.c-i[ciF(s) + c 2 G(s)] = ci.f(x) + c2g(x) = ci .c-i[F(s)] + c2£-i[G(s)].

Thus, we h ave proven the inverse Laplace transform is a linear opera-
tor.

In integral calculus, you learned how to integrate rational functions by using
partial fraction decomposition. In order to find inverse Laplace transforms of
rational functions efficiently using a table of Laplace transforms, we need to
know a variation of partial fraction decomposition.

PARTIAL FRACTION EXPANSION FOR COMPUTING

INVERSE LAPLACE TRANSFORMS

Let F(s) = P(s)/Q(s) where P(s) and Q(s) are polynomials with real

coefficients, where P(s) and Q(s) have no common factor, and where the

degree of P(s) is less than the degree of Q(s).

For Linear Factors

When s - r is a factor of Q(s) exactly n t imes, the part of the partial

fraction expansion for P(s)/Q(s) corresponding to the term (s - rr is

Ai A2 An

--+ s-r (s-r) 2 +···+---(s-r)n

where Ai, A2, ... , An are real constants which must be determined.

For Irreducible Quadratic Factors

Let ( s-a )^2 +b^2 be a quadratic factor of Q( s) which cannot be factored into

linear factors with real coefficients. When (s - a)^2 + b^2 is a factor of Q(s)

exactly n times, the part of the partial fraction expansion for P(s)/Q(s)

corresponding to the term (s - a)^2 + b^2 is

------Bi(s - a)+ Gib + B2(s - a)+ C2b + ... + ------Bn(s - a)+ Cnb

(s - a)2 + b2 ((s - a)2 + b2)2 ((s - a)2 + b2)n

where Bi, B2, ... , Bn and Ci, C2, ... , Cn are real constants which must

be determined.