Math 134 Spring 2022: Final Part 1
ˆ You must show complete work to earn credit. Open book, Open note.
ˆ Do NOT confer with your classmates. Ask me if you have questions.
All parts of the problem refer to the same system. Your goal is to sketch
the phase portrait of the system (use the next page if you’d like) ⃗x˙ = A⃗x
where the matrix A is made up of the last four digits of your student ID
number. For example,if your
ends in 2346, then your
student ID
number
ẋ
2 3
x
system would be ⃗x˙ =
=
= A⃗x.
ẏ
4 6
y
If your matrix has a non-isolated fixed point then you can change one of
the digits. Just write a note explaining why you changed it.
1.
(10 points)
Calculate the eigenvalues and eigenvectors of your matrix.
2. (20 points) Calculate the Nullclines and the straight-line solutions (if
any) and include them on your sketch. Label each line and then draw
arrows to indicate which direction trajectories are moving on that line.
3. (5 points) The Nullclines separate the plane into regions where solutions
are generally moving in the same direction. For each Nullcline-separated
region in your plot, indicate the general direction that solutions are moving
(e.g. “Northeast” or “Down-and-right”).
4. (5 points) Identify all equilibria in the system and classify them as
sources, sinks, saddles, or none of the above. Explain (in your own words)
what this means about solutions that start near that equilibrium.
5. (10 points) Pick any non-equilibrium point on your plane. Explain what
happens to the solution that goes through your point in both forward and
backwards time. Indicate which region(s) of your plane that your solution
moves through (again, in both forward and backwards time).
1.0
0.5
-1.0
0.5
-0.5
-0.5
-1.0
1.0