Review of Analog Signal Processing Basics - Digital Signal Processing: Fundamentals and Applications - page 793

EXAMPLE B.10
Perform the Laplace transform for each of the following functions.
a. xðtÞ¼5 sin ð2tÞuðtÞ
b. xðtÞ¼5e 3t cos ð2tÞuðtÞ

Solution:
a. Using line 5 in Table B.5 and noting that u ¼ 2, the Laplace transform immediately follows:

X ðsÞ¼5Lf2 sin ð2tÞuðtÞg

5 2
s 2 þ 2 2 ¼

10
s 2 þ 4

¼

b. Applying line 9 in Table B.5 with u ¼ 2 and a ¼ 3 yields
X s ¼ 5L e 3t cos 2t u t

5ðs þ 3Þ
ðs þ 3Þ 2 þ2 2 ¼

5ðs þ 3Þ
ðs þ 3Þ 2 þ4

¼

B.2.2 Solving Differential Equations Using the Laplace Transform
One of the important applications of the Laplace transform is to solve differential equations. Using the
differential property in Table B.5 , we can transform a differential equation from the time domain to the
Laplace domain. This will change the differential equation into an algebraic equation, and we then
solve the algebraic equation. Finally, the inverse Laplace operation is processed to yield the time
domain solution.

EXAMPLE B.11
Solve the following differential equation using the Laplace transform:
dy ð t Þ

dt þ 10y t ¼ x t with an initial condition y 0 ¼ 0;

where the input xðtÞ¼5uðtÞ.
Solution:
Applying the Laplace transform on both sides of the differential equation and using the differential property (line
14 in Table B.5 ) , we get

sY ðsÞyð0Þþ10Y ðsÞ¼X ðsÞ

Note that

X s ¼ Lf5uðtÞg ¼ s

Substituting the initial condition yields

Y s ¼

5
sðs þ 10Þ