EEL 3105
Fall 2011
Homework 8
1.
Solve the following problems for both of these the differential equations
2
2
2
2
6
16
( )
12
16
( )
d y
dy
y
u t
dt
dt
d y
dy
y
u t
dt
dt
To get ready to solve the problems below, let us first calculate the roots of the corresponding
polynomials:
2
2
6
16
12
16.
and
The roots are
3
7
j
[labeled as
1
2
,
] and
‐
10.47,
‐
1.53 [labeled as
3
4
,
] respectively. Thus, the
solution the homogeneous part of the first differential equation [I will label it (i)] has the form:
( 3
2.65)
( 3
2.65)
3
[
cos(2.65 )
sin(2.65 )]
j
t
j
t
t
Ae
Be
or
e
M
t
N
t
where A,B or M,N are unknown constants that depend on the initial conditions. Note that this function
is a decaying sinusoid.
Please plot it using MATLAB to understand this important point.
Similarly, the solution to the homogeneous part of the second differential equation [labeled as (ii)] has
the form
10.47
1.53
t
t
Ae
Be
This function, by contrast, has no oscillations but simply decays exponentially to zero. The slower time
constant 1.53 determines, to a dominant extent, the rate of decay to zero. Again, please plot using
MATLAB to get a sense of this function.
We are now ready to attack the problems below.
A.
Suppose y(0)=0,
(0)
0
dy
dt
and u=U(t) where U(t) is the unit step function. Find y(t).

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