Since
I
(0) = 0, then
C
=

V
R
. The solution to the IVP is therefore
I
(
t
) =
V
R
1

e

Rt
L
. The final value is obtained as
t
approaches infinity. Therefore, the final value is
V
R
. We are
looking for
t
?
such that
I
(
t
?
) = 0
.
95
V
R
. This gives
t
?
=
L
R
ln(20)
.
3: MassSpring system
Consider a massspring system. The displacement of the mass from equilibrium is modelled by a
function
x
(
t
). Assuming that the mass is 1 unit, the spring constant is
k >
0, the damping
coefficient is 2
d
where
d >
0, then
x
(
t
) satisfies the ODE:
¨
x
+ 2
d
˙
x
+
kx
= 0
,
2
ECE 205  Winter 2017
Assignment 3 Solutions
Due 05022017
where ˙
x
is the first derivative of the position with respect to time and ¨
x
is the second derivative of
the position with respect to time.
(a)
What choices for
d
yield
underdamped
oscillations? In this regime, what is the angular
frequency of oscillations?
(b)
If the value of
d
is chosen such that the system is on the boundary of being
underdamped
then we say that the system is critically damped. Suppose that the system is critically damped,
and we have the initial values
x
(0) = 1 and ˙
x
(0) =
v
.
Find
the particular solution solving this
IVP in terms of
v
.
Show
that no matter what the initial velocity
v
is, the mass can never pass
the equilibrium point more than once.
4: Almost Simple Harmonic Oscillator
Consider the ODE
y
00
(
t
) + 2
λy
0
(
t
) +
ω
2
0
y
(
t
) = 0
.
How many oscillations take place until the amplitude decays by 1% when: