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PRELIM
2
, MATH
293
, SPRING
2006
STUDENT’S NAME:
TA’S NAME:
PROBLEM 1:
PROBLEM 2:
PROBLEM 3:
PROBLEM 4:
PROBLEM 5:
TOTAL:
1
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PRELIM 2, MATH 293, SPRING 2006
Problem 0.1.
Consider the diﬀerential equation
dx
dt
=
rx
+
x
3
containing the parameter
r
.
(
a
)
Calculate all the critical points for all the values of
r
and study their stability.
(
b
)
Find the bifurcation point.
(
c
)
Sketch the bifurcation diagram.
Proof
(a) Let
f
(
x
) =
rx
+
x
3
=
x
(
x
2
+
r
). The critical points are the real zeros of
f
. A
critical point
x
is stable if and only if
dx
dt
is increasing just to the left of
x
and
decreasing just to the right of
x
, or, equivalently,
f
0
(
x
)
<
0. (Why don’t they tell
you this in the book?) Notice that
f
0
(
x
) =
r
+ 3
x
2
.
If
r
≥
0 then the only critical point is at
x
= 0, and it is unstable because
f
0
(0) =
r
≥
0.
If
r <
0, the critical points are at
x
= 0
,
√

r,

√

r
. The critical point at
x
= 0 is stable because
f
0
(0) =
r <
0. The critical points at
x
=
±
√

r
are
unstable because
f
0
(
±
√

r
) =

2
r >
0.
(b) The bifurcation point is the value of
r
at which the number of critical points
changes. The bifurcation point is at
r
= 0.
(c) The bifurcation diagram is a plot of the critical points against the parameter,
namely
PRELIM 2, MATH 293, SPRING 2006
3
Problem 0.2.
A jet ski is traveling across the water with an initial constant velocity of
v
0
. At time
t
= 0
, the jet skier cranks the throttle, turning on the motor, which provides
a constant acceleration of
a
2
m/
s
2
, but the water resistance causes the jet ski to decelerate
at
ρ
2
v
2
m/
s
2
, where the velocity is
v
m/s.
(
a
)
Draw a diagram labeling the direction, velocity and acceleration vectors of the jet
ski, as well as the direction vector of the water resistance (i.e., free body diagram).
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This homework help was uploaded on 01/21/2008 for the course MATH 2930 taught by Professor Terrell,r during the Spring '07 term at Cornell University (Engineering School).
 Spring '07
 TERRELL,R
 Math

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