MATH 1280 HOMEWORK 11 SOLUTIONS
Section 8.3
Problem 1: Well be investigating the Brusselator in this problem. We take a, b > 0 as parameters and
x, y 0 as dimensionless concentrations.
x = 1 (b + 1)x + ax2 y
y = bx ax2 y
Solution:
Part (a): To nd and clas
MATH 1280 HOMEWORK 4 SOLUTIONS
Section 4.1
Problem 2: Find and classify the xed points and sketch the phase portrait on the circle for = 1 + 2 cos .
1
2 4
Solution: Fixed points occur when = 0, which gives cos = , or =
,
. We can gure out the
2
3 3
stabil
MATH 1280 HOMEWORK 1 SOLUTIONS
Section 2.2
14
Problem 2: Graphically analyze x = 1 x , then try to nd an analytic solution.
Solution: First, we nd the equilibrium points by looking at x = 0. There are only two real solutions,
x = 1, x = 1. Next, we plot x
MATH 1280 HOMEWORK 7 SOLUTIONS
Section 6.4
Problem 2: Analyze the system x = x(3 2x y), y = y(2 x y).
Solution: The Jacobian for this system is
J(x, y) =
3 4x y
y
x
2 x 2y
There are 4 possible xed points, which we classify below.
Fixed point (0, 0): unsta
MATH 1280 HOMEWORK 6 SOLUTIONS
Section 6.1
Problem 4: Find and classify the xed points. Sketch the nullclines, the vector eld, and a plausible phase
portrait. x = y, y = x(1 + y) 1.
Solution: There is a single xed point at (1, 0). The Jacobian is given by
MATH 1280 HOMEWORK 9 SOLUTIONS
Section 7.3
Problem 1: Consider x = x y x(x2 + 5y 2 ), y = x + y y(x2 + y 2 ).
Solution:
1 1
Part (a): The Jacobian at the origin is given by
. The trace and the determinant are both
1 1
positive, and since 2 4 < 0, we have
MATH 1280 HOMEWORK 10 SOLUTIONS
Section 8.1
Problem 5: Prove that at any zero-eigenvalue bifurcation in two dimensions, the nullclines always intersect
tangentially.
Solution: Let our system be given by x = f (x, y), y = g(x, y). The x-nullcline is dened
MATH 1280 HOMEWORK 2 SOLUTIONS
Section 2.5
Problem 5: Consider the initial value problem x = |x|p/q , x(0) = 0, where p and q are positive integraers
with no common factors.
(a) Show that there are an innite number of solutions if p < q.
(b) Show that the
MATH 1280 HOMEWORK 3 SOLUTIONS
Section 3.4
Problem 2: Collect the bifurcation information for x = rx sinh x.
Solution: Fixed points occur at intersections of rx and sinh x. The three qualitatively dierent cases are
r < 1, r = 1, r > 1. Cartoons of rx and
MATH 1280 HOMEWORK 8 SOLUTIONS
Section 7.1
Problem 2: Sketch the phase portrait for r = r(1 r2 )(9 r2 ), = 1.
Solution:
From the form of r, we see that there are two radii that qualify as periodic orbits: r = 1 and r = 3. r = 3 is
unstable and r = 1 is st
MATH 1280 HOMEWORK 5 SOLUTIONS
Section 5.1
Problem 9: Consider the system x = y, y = x.
Solution:
Part a:
y
x
Part b: We note that xx y y = xy (xy) = 0, so we can integrate both sides directly, or just observe
d 2
x y 2 = 2xx 2y y = 0 x2 y 2 = C
dt
Since
Week 4 - REVISED
Winter/Spring 2011
Math 1280 Ordinary Dierential Equations 2 - RUBIN -
SCHEDULE: Homework from this handout is due at the start of class on Monday, January
31st.
Monday, January 24th: start Section 3.1
Wednesday, January 26th: nish Sect
MID-SEMESTER EVALUATION
MATH 1280 - J. Rubin
March 14, 2011
Please answer the following questions to provide feedback that I can use in this course. This is not
an ocial evaluation form, and I will read the answers ASAP.
Circle the most appropriate choice