Page 116 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

106 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
Taking the Laplace transform

1 1
2
(s − 1)Y = −
s s − 3
1 1 1 1
Y (s ) = − = −
2
2
s (s − 1) (s − 3)(s − 1) s (s + 1)(s − 1) (s − 3)(s + 1)(s − 1)
First find the inverse transform of the first term.

3
q = s − s
2
q = 3s − 1

q (0) =−1 p(0) = 1
q (1) = 2 p(1) = 1

q (−1) = 2 p(−1) = 1

The inverse transform is

t
−1 + 1/2e + 1/2 e −t

Next consider the second term.

3
2
q = s − 3s − s + 3
2
q = 3s − 6s − 1

q (−3) = 44 p(−3) = 1

q (1) =−4 p(1) = 1

q (−1) = 8 p(−1) = 1

The inverse transform is

1 1 1
t
e −3t − e + e −t
44 4 8
Thus

1 t 5 −t 1 −3t
y(t) = e + e + e − 1
4 8 44