CHAP. 16]

THE LAPLACE TRANSFORM METHOD

Another Method  ðt  1=ðs þ aÞ a a ¼ l a e d ¼ a s sðs þ aÞ 0

16.4

Find l1



1 2 sðs  a2 Þ



Using the method of partial fractions, 1 A B C þ ¼ þ sðs2  a2 Þ s s þ a s  a and the coeﬃcients are A¼ Hence,

   1  1 1  1 1  1 ¼  B ¼ ¼ C ¼ ¼ sðs  aÞ s¼a 2a2 sðs þ aÞ s¼a 2a2 s2  a2 s¼0 a2 " # " # " #   2 2 2 1 1 1=a 1 1=2a 1 1=2a ¼ l l1 þ l þ l s sþa sa sðs2  a2 Þ

The corresponding time functions are found in Table 16-1:   1 1 1 1 1 ¼  2 þ 2 eat þ 2 eat l sðs2  a2 Þ a 2s 2a   1 1 eat þ eat 1 ¼ 2 ðcosh at  1Þ ¼ 2þ 2 2 a a a Another Method By lines 11 and 14 of Table 16-1, " # ð   2 2 t sinh a cosh a t 1 1 1=ðs  a Þ l ¼ 2 ðcosh at  1Þ ¼ d ¼ s a a2 a 0 0

16.5

Find l

1



sþ1 sðs2 þ 4s þ 4Þ



Using the method of partial fractions, we have

Then and Hence,

sþ1 A B B2 ¼ þ 1 þ s s þ 2 ðs þ 2Þ2 sðs þ 2Þ2   s þ 1  1 s þ 1  1 A¼ ¼ ¼ ¼ B 2 s s¼2 2 ðs þ 2Þ2 s¼0 4  s þ 2  1 B1 ¼ ðs þ 2Þ ¼ 4 2sðs þ 2Þ2 s¼2 1  1     1  sþ1 1 1 4 1  4 1 2 þl þl ¼l l s sþ2 sðs2 þ 4s þ 4Þ ðs þ 2Þ2

The corresponding time functions are found in Table 16-1:   sþ1 1 1 1 l1 ¼  e2t þ te2t 4 4 2 sðs2 þ 4s þ 4Þ

407

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An
Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An