Page 118 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

108 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
Differentiate once
1

φ = φ (−2) = 1 = A 1
(s + 1) 2
−2

φ = φ (−2) = 2 = A 0
(s + 1) 3
To find C,multiplyby(s + 1) and take s =−1(in theoriginalequation).

C =−1.

Thus
2 1 2 1
F(s ) = + + −
(s + 2) (s + 2) 2 (s + 2) 3 (s + 1)
m
and noting the shifting theorem and the theorem on t ,
2 −2t
f (t) = 2e −2t + te −2t + 2t e + e −t

6.5.3 Quadratic Factors: Complex Roots
If q(s ) has complex roots and all the coefficients are real this part of q(s ) can always be written
in the form
2
(s − a) + b 2 (6.62)
This is a shifted form of

2
s + b 2 (6.63)
This factor in the denominator leads to sines or cosines.

Example 6.9. Quadratic factors
Find the inverse transform of

2(s − 1) 2s 1
F(s ) = = −
2
2
2
s + 2s + 5 (s + 1) + 4 (s + 1) + 4
Because of the shifted s in the denominator the numerator of the first term must also be shifted
to be consistent. Thus we rewrite as
2(s + 1) 3
F(s ) = −
2
2
(s + 1) + 4 (s + 1) + 4
The inverse transform of
2s
2
s + 4