Lecture Notes in Differential Equations - Bruce E. Shapiro

354 LESSON 32. THE LAPLACE TRANSFORM
The inverse transform is
[ ] [ ] [ ]
y(t) = 2 · L −1 s
9 + s 2 + 5 · L −1 1
9 + s 2 + L −1 s
(9 + s 2 ) 2
= 2 3 cos 3t + 5 [ ] [ ]
3 sin 3t + s
L−1
9 + s 2 ∗ L −1 1
9 + s 2
(32.213)
(32.214)
= 2 3 cos 3t + 5 3 sin 3t + 1 cos 3t ∗ sin 3t (32.215)
9
The convolution is
hence
cos 3t ∗ sin 3t =
∫ t
0
cos 3x sin 3(t − x)dx (32.216)
= 1 t sin(3t) (32.217)
2
y(t) = 2 3 cos 3t + 5 3 sin 3t + 1 t sin 3t (32.218)
18
Impulses
Impulses are short, sudden perturbations of a system: quickly tapping on
the accelerator of your car, flicking a light switch on and off, injection of
medicine into the bloodstream, etc. It is convenient to describe these by
box functions – with a step on followed by a step off. We define the Unit
Impulse of Width a at the Origin by the function
⎧
⎪⎨ 0 t < −a
δ a (t) = 2a −a ≤ t < a
(32.219)
⎪⎩
0 t > a
In terms of the unit step function,
δ a (t) = 1 (U(t + a) − U(t − a)) (32.220)
2a
As the value of a is decreased the width of the box gets narrower but
the height increases, making it much more of a sudden spike, but in each
case, the area of the box is unity. In the limit, a sequence of narrower
and narrower boxes approaches an infinitely tall spike which we call the
Dirac-delta function 2
δ(t) = lim δ a (t) (32.221)
a→0
2 For Paul Dirac (1902-1982), one of the founders of quantum mechanics.