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 p
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 o
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.
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 = ?
So
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
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 met
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).
So
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)
pa
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 becaus
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 )
=
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 (
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 be