Solutions Final Exam
1. (20 points) Determine a 3 3 linear system of ODEs whose phase portrait
contains an unstable line x1 = 2x3 , x2 = 0; and
contains the circle given by the intersection of the plane x1 + x2 = 0 with the unit sphere
x2 + x2 + x2 = 1.
Homework 7 Sample Solutions
Problem #5. Deterime the asymptotic behavior of the following IVP as t
x + x = est + 2et + t2
x(0) = 0, x (0) = 0, x (0) = 0
Solution. This solution also serves as a bit of an apology; I didnt cover this technique in
section,
Homework 2 Sample Solutions
Problem 2.3. In Figure 2.2 (not pictured here), you see four direction elds. Match each
of these direction elds with one of the systems in the previous exercise.
Solution. Many of you just matched the pictures with the systems,
Homework 4 Sample Solutions
Problem 4.6. Prove that any two linear systems with the same eigenvalues i , = 0 are
conjugate. What happens if the systems have eigenvalues i and i with = ? What
if = ?
Solution. People actually generally got this one right, b
Homework 5 Sample Solutions
Problem 6.12. Compute the exponentials of the following matrices. (Listed below)
Solution. Im not going to do all of these, but Ill pick a representative sample.
5 6
(a) A =
3 4
With matrices like this one, its easiest to rst m
Homework 6 Sample Solutions
Problem #1. Find a particular solution of the following second order ODEs:
a) x + x = et
b) x x = et
Solution. Several people used a complicated variation of parameters method to solve these,
so I thought Id illustrate how much
Homework 3 Sample Solutions
Problem 3.1. In Figure 3.9 (not pictured here), you see six phase portraits. Match each
of these phase portraits with one of the following linear systems (listed below).
Solution. Notice that each of the given diagrams has noti
Homework 8 Sample Solutions
Problem #6. Let F : Rn Rn be C 1 . Suppose that the autonomous system X = F (X)
admits a global solution X(t) with X(t) = C cos tE1 + C sin tE2 for some C > 0, i.e. X(t)
parameterizes a circle.
(a) Show that if n = 2 and |X0 |
Homework 1 Sample Solutions
Problem 1.4(b). Let ga (x) = f (x) + a. Sketch the bifurcation diagram corresponding to
the family of dierential equations x = ga (x).
Solution. I chose this problem because most students gave no indication of how they found
th
Solutions Midterm Exam 1
1. Find all solutions y = y(x) to the following initial value problems (remember to include domain):
(a) (10 points)
x2 y = y(1 + y 2 )
y(0) = 0
(Hint: use that
1
y(1+y 2 )
=
1
y
y
).
y 2 +1
We rst observe that this ODE is poorly
Solutions Midterm Exam 2
1. Consider the forced 2 2 linear system
3et
5et
Y = AY +
where
A=
14 9
.
25 16
(a) (10 points) Compute the matrix exponential etA .
The characteristic polynomial of A is pA () = 2 2 + 1. Hence, the there is one eigenvalue
= 1 wi
Homework 9 Sample Solutions
Problem 8.5. Consider the system
x = x2 + y
y =xy+a
where a is a parameter.
(a) Find all equilibrium points and compute the linearized equation at each.
(b) Describe the behavior of the linearized system at eacu equilibrium poi