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Unformatted text preview: um=[2 T 4 ]; d en=[eonv(eonv([1 1 ],[1 2 ]),[1 2J)J;
[r, p , k ]= r esidue(num,den); = 3, r z [ Hence, 2, -1 = and f (t) 2t = ( 3e- + 2 te- 2t - e-t)u(t) ( c) r = 3 .5 - 0.4811i. 3 .5 + 0 .4811i. 1 .00
P = - 0.5 + 2 .5981i. - 0.5 - 2.5981i. - 2.00
a ngle = - 0.1366. 0 .1366. 0
mag = 3 .5329. 3 .5329. 1 .00
F (s) = - s +2 3.5329 e -jO.1366 + s + 0.5 - j2.5981 3.5329 e jO.1366 + --=-=-,-::-::c~
8 + 0.5 + j 2.5981 and
f (t) = [ e- 2t o E xercise E 6.2
(i) Show that the Laplace transform of l Oe- 3t cos (4t lOa from Table 6.1.
( ii) Find the inverse Laplace transform of: (a) ( e) [3e 2t 321
s + 2 + (8 + 2)2 - s + 1 n um=[8 2 1 1 9]; d en=[conv([O 1 2 ],[1 1 7])];
[r, p , k j= r esidue(num,den)
[ angle,mag] = eart2pol(real(r) , imag(r» Hence, 379 82 + 53.13°) is 6s - 14 .
82 + 68 + 25 Use Pair + 17 ( b)
38 - 5
( 8+ 1 )(82 + 2s+5) 8 Answers:(a) (3e t -2e- 5t )u(t) ( b) [-2e-t+~e-tcos(2t-36.87°)Ju(t) 1
F (s) T he Laplace Transform 168 + 43
( c) (8 _ 2)(8 + 3)2 P = - 2. - 2. - 1
k 6.1 + l .766e-O. 5 tcos(2.598lt - 0.1366)Ju(t) 0 C omputer E xample C 6.2 Find (a) t he direct Laplace transform of sin a t + cos bt (b) the inverse Laplace
transform of (as 2)/(s2 + b2).
Here we shall use S ymbolic Math Toolbox, which is a collection of functions for
MATLAB used for manipulating and solving symbolic expressions.
F =laplace(f) F =(a*s-2+b-2*a+s-3+s*a-2)/(s-2+a-2)/(s-2+b-2) Thus, ( b)
F =sym('( a *s- 2)/(s- 2+b- 2);
F =a*dirac(t)-a*b*sin(b*t) Thus,
f (t) = a6(t) + absin(bt)u(t) 0 + (t - 3)e- 3t Ju (t) \l A Historical Note: Marquis Pierre-Simon De Laplace ( 1749-1827)
T he Laplace transform is named after t he g reat French mathematician a nd
a stronomer Laplace, who first presented t he t ransform a nd its applications to differential equations in a p aper published in 1779.
Laplace developed t he foundations of potential theory a nd m ade i mportant
c ontributions t o special functions, probability theory, astronomy, a nd celestial mechanics. In his E xposition du s ysteme du Monde (1796), Laplace formulated a
nebular hypothesis o f cosmic origin a nd t ried t o explain t he universe as a pure
mechanism. In his Traite de M echanique Celeste (celestial mechanics), which completed t he work of Newton, Laplace used mathematics a nd physics t o s ubject t he
s olar system a nd all heavenly bodies t o t he laws o f m otion a nd t he principle of
gravitation. Newton had been unable to explain t he irregularities of some heavenly
bodies; in desperation, he concluded t hat God himself must intervene now a nd t hen
t o prevent some catastrophes, such as J upiter eventually falling into t he s un (and
t he moon into t he e arth) as predicted by Newton's calculations. Laplace proposed
t o show t hat these irregularities would correct themselves periodically, a nd t hat a
l ittle p atience-in J upiter's case, 929 y ears-would see everything returning a utomatically t o order, a nd t here was no reason why t he solar a nd t he s tellar systems
could not continue to...
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