Math 2326
Practice Exam 2
There will be 6-8 problems on the actual test. All of them will be similar to the problems shown here.
1) Use the guess and test method to find two solutions
and
to the second-order differential equation for the damped harmonic
oscillator represented by
(The solutions are
and
)
2) Consider the partially-decoupled system
a) Find the general solution to this system.
b) Find the solution to the system with initial-value
(The general solution is
problem is
. The solution to the initial value
.)
3) At right is a phase portrait for the logistic predator-prey system
Match each initial condition
and on the phase portrait to its
- and
-graph. Choose from the four graphs below. One of the
choices does not match any of the initial conditions given on the phase
portrait.
( matches the upper left graph, matches the upper right graph, and
matches the lower right graph)
R, F
R, F
2.5
3.0
2.0
2.5
2.0
1.5
1.5
1.0
1.0
0.5
0.5
t
0.0
0
2
4
6
8
10
R, F
t
0.0
0
12
2
4
6
8
10
12
R, F
3.0
2.0
2.5
1.5
2.0
1.5
1.0
1.0
0.5
0.5
t
0.0
0
2
4
6
8
10
12
t
0.0
0
2
4
6
8
10
12
4) Consider the second-order differential equation
a) Convert this equation to a first-order system of equations by letting
(The system is
and
.
)
5) Consider the system
a) Find the general solution for this system.
b) Find the solution for the system if the initial condition is
(The general solution is
and
. Note also that any multiple of the vectors
is also acceptable. The solution of the initial-value problem is
6) Find all equilibrium points for the predator-prey system
(Equilibrium points are
and
)
7) Consider the system
a) Find the general solution.
(General solution is
)
b) Find the particular solution with the initial value
.
(Particular solution is
)
c) Does this system represent a source, a sink, a spiral source, a spiral sink, or a center?
(It’s a spiral source)
8) Consider the system
a) Find the general solution.
(General solution is
b) Find the particular solution with the initial value
(Particular solution is
)
.
)
c) Does this system represent a source, a sink, a spiral source, a spiral sink, or a center?
(It’s a sink)
9) Consider the second-order differential equation
This equation models a harmonic oscillator.
a) Find the general solution to the equation. (Find both
(Solution is
and
).
)
b) Find the solution if the initial value is
(Solution is
)
c) Is this oscillator undamped, underdamped, overdamped, or critically damped?
(It’s overdamped)
10) Consider the second-order differential equation
This equation models a harmonic oscillator.
a) Find the general solution to the equation. (Find both
General solution is
and
).
b) Find the solution if the initial value is
(Particular solution is
)
c) Is this oscillator undamped, underdamped, overdamped, or critically damped?
(It’s underdamped)