THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

640 Appendix C. Using Laplace Transforms to Solve Differential EquationsThe residues C 1 , C 2 , and C 3 are obtained from Equation (C.11) as follows:s + 3C 1 = lims→0 (s + 1)(s + 2) = 3(1)(2) = 3 2s + 3C 2 = lims→−1 s(s + 2) = 2(−1)(1) =−2s + 3C 3 = lims→−2 s(s + 1) = 1(−2)(−1) = 1 2Hence( 3/2X(s) = 13 + −2s s + 1 + 1/2 )s + 2Applying the transform pairs we derived in Section C.2,( 3x(t) = 132 − 2e−t + 1 )2 e−2tWhen we know the form of the inverse transform above we can derive the residuegraphically by inspection. The method for doing this is simply divide the productof all vectors from the zeros to the pole whose residue is being determined by theproduct of the vectors from all the other poles to the pole under consideration. Thistechnique is shown in Figure C.1. In Figure C.1(a), C 1 is derived as3̸0 ◦C 1 =(2̸ 0 ◦ )(1̸ 0 ◦ ) = 3/2In Figure C.1(b), we get2̸ 0 ◦C 2 =(1̸ 0 ◦ )(1̸ 180 ◦ ) = 2(1)(−1) =−2and from Figure C.1(c):1̸ 0 ◦C 3 =(1̸ 180 ◦ )(2̸ 180 ◦ ) = 1(−1)(−2) = 1/2Note that the numbers obtained from the use of the graphical residue techniqueare the same as those derived from Equation (C.11). The sign of the residue can bereadily established by simply counting the number of poles and zeros to the rightin the s plane of the pole whose residue is being determined. If the number is even,the sign of the residue is positive; and if the number is odd, the sign is negative.Example Problem 2Consider the function8(s + 2)(s + 4)X(s) =s(s + 1)(s + 3)Find the inverse Laplace transform.