Page 241 - Schaum's Outline of Differential Equations

CHAPTER 22

Inverse Laplace

Transforms

DEFINITION
t
An inverse Laplace transform of A'(.v), designated by *f~ {F(s)}, is another function J(x) having the prop-
erty that S{/U)} = F(.v). This presumes that the independent \ariahle of interest is .v. If the independent vari-
able of interest is / instead, then an inverse Laplace transform of Fix) if fit) where -£{.fti)} = F(s).
The simplest technique for identifying inverse Laplace transforms i.s to recogni/.e them, either from mem-
orv or from a tahle such as Appendix A (see Problems 22.1 through 22.3). If F(x) is not in a recognisable form,
then occasionally it can be transformed into such a form in algebraic manipulation. Observe from Appendix A
that almost all Laplace transforms are quotients. The recommended procedure is to first convert the denominator
to a form that appears in Appendix A and then the numerator.

MANIPULATING DENOMINATORS
The method of completing the square converts a quadratic po!\ normal into the sum of squares, a form that
appears in many of the denominators in Appendix A. Iti particular, for the quadratic as^+bs + c. where a.b, and
c denote constants.

where A- = b!2a and (See Problems 22.8 through 22.10.)
The method of partial fractions transforms a function of the form a(x)/b(s). where both a(s) and b(s) are
polynomials in .v. into the sum of other fractions such that the denominator of each new fraction is either a first-
degree or a quadratic polynomial raised to some power. The method requires onl\ that I) the degree of a(s) be
(
less than the degree of/Xs) (if this is not the ease, first perform long division, and consider the remainder term)
and (2) b(x) he factored into the product of distinct linear and quadratic polynomials raised to \arious powers.

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