Page 112 - Essentials of applied mathematics for scientists and engineers
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book Mobk070 March 22, 2007 11:7
102 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
so
∞
dF −st
=− tf (t)e dt (6.32)
ds
t=0
and in general
n
d F
n
= L[(−t) f (t)] (6.33)
ds n
For example
d k 2sk
L[t sin(kt)] =− = (6.34)
2
ds s + k 2 (s + k )
2 2
2
6.3 LINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH
CONSTANT COEFFICIENTS
Example 6.5. A homogeneous linear ordinary differential equation
Consider the differential equation
y + 4y + 3y = 0
y(0) = 0 (6.35)
y (0) = 2
2
2
L[y ] = s Y − sy(0) − y (0) = s Y − 2 (6.36)
L[y ] = sY − y(0) = s Y (6.37)
Therefore
2
(s + 4s + 3)Y = 2 (6.38)
2 A B
Y = = + (6.39)
(s + 1)(s + 3) s + 1 s + 3
To solve for A and B, note that clearing fractions,
A(s + 3) + B(s + 1) 2
= (6.40)
(s + 1)(s + 3) (s + 1)(s + 3)
Equating the numerators, or
A + B = 03A + B = 2: A = 1 B =−1 (6.41)
