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

642 Appendix C. Using Laplace Transforms to Solve Differential EquationsFigure C.2. Pole-zero pattern for Example Problem 2.The partial fraction expansion of the function must be of the formatA(s)B(s) = C 11(s + a 1 ) m + C 12(s + a 1 ) m−1 +··· C 1ms + a 1+ C 2s + a 2+···+C ns + a nThe residues C 2 , C 3 ,...,C n are determined in the same manner as was describedin the preceding section but the coefficients C 11 , C 12 ,..., C 1m require specialtreatment. They are evaluated from the following expression:1C 1 j =( j − 1)!d j−1ds j−1 (s + a 1) m A(s)B(s)(C.12)which is evaluated at s =−a 1 . Then the inverse Laplace is obtained from thetransform relation:[ ]L −1 1= t j+1(s + a) j ( j − 1)! e−at (C.13)The following example problem will demonstrate the procedure.Example Problem 3Find the inverse Laplace transform forX(s) =s + 1(s + 2) 2 (s + 3) = C 11(s + 2) 2 + C 12s + 2 + C 2s + 3