Page 242 - Schaum's Outline of Differential Equations

CHAP. 22] INVERSE LAPLACE TRANSFORMS 225

m
The method is carried out as follows. To each factor of b(s) of the form (s - a) , assign a sum of m frac-
tions, of the form

p
To each factor of b(s) of the form (s 2 + bs + c) , assign a sum of p fractions, of the form

Here A;, Bj, and C k(i= 1, 2,..., m;j, k= 1, 2,...,p) are constants which still must be determined.
Set the original fraction a(s)lb(s) equal to the sum of the new fractions just constructed. Clear the resulting
equation of fractions and then equate coefficients of like powers of s, thereby obtaining a set of simultaneous
linear equations in the unknown constants A t, Bj, and C k. Finally, solve these equations for A t, Bj, and C k. (See
Problems 22.11 through 22.14.)

MANIPULATING NUMERATORS
A factor s - a in the numerators may be written in terms of the factor s - b, where both a and b are
constants, through the identity s - a = (s - b) + (b - a). The multiplicative constant a in the numerator may be
written explicitly in terms of the multiplicative constant b through the identity

Both identities generate recognizable inverse Laplace transforms when they are combined with:
Property 22.1. (Linearity). If the inverse Laplace transforms of two functions F(s) and G(s) exist, then
for any constants c l and c 2,

(See Problems 22.4 through 22.7.)

Solved Problems

22.1. Find

Here F(s) = Us. From either Problem 21.4 or entry 1 of Appendix A, we have ^{1} = Us. Therefore,
l
£- {lls} = 1.

22.2. Find

From either Problem 21.6 or entry 7 of Appendix A with a = 8, we have

Therefore,