ACM 95b/100b
Solutions for Problem Set 2
Sean Mauch
[email protected]
January 18, 2002
1
Problem 1
Problem.
Four ladybugs start at the corners of a unit square (
±
1
2
±
ı
2
). At
t
0
, each beetle moves
along a smooth trajectory that is continuously adjusted to point toward the next beetle along the
square in a counterclockwise direction. At any given time, all four beetles are moving at the same
speed.
1. Using complex dependent variables to simplify your formulation, write down a system of four
differential equations with initial conditions that governs the motion of the beetles as they
spiral in to meet at the origin.
2. Use the symmetry of the problem to replace this system with a single first order initial value
problem that describes the motion of one of the beetles along its trajectory.
3. Solve this IVP and sketch the solution.
How many times does the beetle spiral around the
origin?
4. Determine the length of this trajectory from
t
0
until the time at which the beetles meet at the
origin.
Solution.
1. Let
z
k
be the position of the beetle that is initially in quadrant
k
+ 1.
We write down the
system of differential equations that govern their motion. We choose a speed that simplifies
the equations namely, the speed is the distance to the adjacent beetle.
z
0
0
=
z
1

z
0
z
0
1
=
z
2

z
0
z
0
2
=
z
3

z
2
z
0
3
=
z
0

z
3
The initial conditions are
z
k
(
t
0
) =
e
ı
(2
k
+1)
π/
4
√
2
.
2. We note that the beetles are each separated by an angle of
π/
2.
The position of beetle
k
+ 1 mod 4 is
ı
times the position of beetle
k
.
This observation allows us to reduce the
problem to a single initial value problem for beetle zero. We take the start time to be
t
0
= 0.
z
0
=
ız

z,
z
(0) =
1 +
ı
2
The position of beetle
k
is
z
k
=
ı
k
z
.
1
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3. We solve this initial value problem. The general solution of the differential equation is
z
=
c
e
(
ı

1)
t
.
We determine the constant of integration from the initial condition.
z
=
e
(
ı

1)
t
+
ıπ/
4
√
2
The angle of the beetle is
t
+
π/
4. Its radius is e

t
. Thus the beetle spirals around the origin
an infinite number of times. The four beetle paths are shown in Figure 1.
Figure 1: The paths of the four beetles.
4. We integrate to determine the length of the trajectory.
L
=
Z
∞
0

z
0

d
t
=
Z
∞
0
(
ı

1) e
(
ı

1)
t
+
ıπ/
4
√
2
d
t
=
Z
∞
0
e

t
d
t
= 1
The trajectory has unit length.
2
Problem 2
Problem.
For the following linear 2
nd
order nonhomogeneous ODE’s, state the intervals in which
a unique solution satisfying initial conditions
y
(
x
0
) =
y
0
, y
0
(
x
0
) =
y
0
0
is guaranteed to exist, where
x
0
is any point in the interval.
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 Winter '09
 NilesA.Pierce
 general solution

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