Fall-11. Math 123 (ccn 54356):
Ordinary Differential Equations
Instructor: Alexander Givental
MWF 11-12, 4 Evans
Office hours: M 1-3 p.m., in 701 Evans
Text: Vladimir I. Arnold, Ordinary Differential Equations,
Springer (2st English translation, yellow cover).
We will "read along" Chapters 1, 2, as much as possible of Chapter 3, and maybe
something from Chapter 4.
Grading: Final (40%)+Midterm (20%)+Homework (40%)
HW: Weekly homework assignments are posted
to this web-page, and your solutions are due on Fri in class.
Typically your homework will be returned to you in a week from the due date
with some problems graded.
Midterm: How about Friday, October 7?
Final: Monday, Dec 12, 11:30-2:30.
HW1, due by Fri, Sep 2: Read sections 1.1-1.12. Solve:
1. Problem 2 from Section 1.8.
2. Draw the vector field (on the line) and direction field (on the
plane) of the differential equation dx/dt = x^2 (1-x)
3. Find out for which values of n=0,1,2,..., all solutions of the
differential equation dx/dt = x^n extend indefinitely in time?
HW2, due by Fri, Sep 9: Read the end of Section 1 and Section 2.
Solve: Problems 4, 5 of Section 1.15, and Problem 2 of Section 1.18.
HW3, due by Fri, Sep 16: Read Sections 2 and 3.
Solve: Problem 1 of Section 2.6, Problem 1 of Section 2.7, and Problem 1 of
HW4, due by Fri, Sep 23: Read Section 4, start 5. Solve:
(i) Problem 1 of Section 3.5.
(ii) Compute the phase flow of the
differential equation y''=-ky (the 2nd derivative of y in x is equal to -ky).
(iii) Prove that the differential
equation dy/dx=y^3 (y cubed) does not define a flow on the line, and that
dy/dx=y^3 cos(y) does.
HW5, due by Fri, Sep 30: Read Section 6. Solve:
Problem 2 of Section 5.4 (an answer is given, but you need to actually find
the transformed equation)
Problem 3 of Section 5.5 (the answer is given, but you need to justify
it - in both cases: when the answer is "yes" and when it is "no"), and
Problem 1 of Section 6.2.
HW6, due by Fri, Oct 7: Read Section 7. Solve:
Problems 2, 3, 4 of Section 6.5.
HW7, due by Fri, Oct. 14: Read Sections 7 and 8. Solve:
Problem 6 of Section 7.1, and Problem 2 (2),(3) of Section 7.7.
HW8, due by Fri, Oct. 21: Read Section 10. Solve:
(i) Problem 1 of Section 10.3,
(ii) Compute pairwise Poisson brackets of vector fields x^k d/dx on the line,
(iii) Problem 1 of Section 10.4,
HW9, due by Fri, Oct. 28: Read Section 11. Solve:
Problem 8 of Section 10.4, Problem 3 of Section 11.1 (the answer is given, but
you need to fully justify it), and Problem 8 of Section 11.7
HW10, due by Fri, Nov. 4: Read Sections 12, 13. Solve:
(i) Problem 2 of Section 12.3,
(ii) Let S(E) denote the area enclosed by an oval on the phase plane given by
Total Energy (x_1, x_2) =E.
When the energy level E increases, the oval changes, and the
area increases. Prove that the period of revolution of a phase point along
the oval with the energy E is equal to the derivative dS(E)/dE of the area
function. Hint: Read top of p. 147.
(iii) Problem 1 of Section 13.1 (invariance of linearization near an
equilibrium). Also, prove that the linear system described by the
formulas of Definition on p. 153 is never independent of the choice of
arbitrary coordinate systems, if the point (x=0) of linearization is not an
equilibrium of the original (non-linear) vector field. Hint: Apply
rectification theorem for vector fields.
HW11, due by Mon, Nov. 14:
(i) Problem 2 of Section 13.2.
(ii) Problem 3 of Section 14.9.
(iii) Problem 1 of Section 15.1.
HW12, due by Fri, Nov. 18:
(i) Compute e^A (e to the power A), where A is the 2x2-matrix with the first row [a,b], and the second row [-b,a].
(ii) In the 3-dimensional phase space of the differential equation
x'''-3x''+5x'-7x=0, consider the region U(t) (depending on t) which is formed
by points (a,b,c)=(x(t), x'(t), x''(t)), where t --> x(t) is any solution
with the initial condidtion lying in the unit cube: 0 < x(0), x'(0), x''(0)< 1.
Find the volume of U(t).
(iii) Find a general formula for the terms of the sequence 1,3,4,7,11,18, ...
where each next term is the sum of the previous two.
HW13, due by Mon, Nov. 28: Read sections 20-21. Solve:
(i) Problem 2 of Section 18.4,
(ii) Problem 1 of Section 20.6,
(iii) Find the solution of the system: dx/dt = iy, dy/dt=-ix, satisfying the
initial condition: x(0)=0, y(0)=1+i.
HW14, due by Fr, Dec. 2: Read sections 22-23. Solve: Problem 1 of
Section 21.3, Problem 1 of Section 22.5.
HW15, due by Fri, Dec. 9: Answer the following question. Let
(i) x'=v(x) and (ii) x'=Ax
be respectively: a (possibly non-linear) vector field in R^n with an
equilibrium at the origin, and the linear vector field obtained as
linearization of (i) at the origin. An equilibrium x=0
can be Lyapunov-unstable, Lyapunov-stable but asymptotically unstable, or
asymptotically (and hence Lyapunov-) stable. This trichotomy applies to each,
the equilibria: of (i) and of (ii), so a priori there are 9
possible combinations of verdicts about the stability of an equilibrium and
Find out which of the 9 combinations are possible, i.e. for those which are
possible, give an example, and for those which are not possible, prove that
they are not. For example, I claim that it is possible that (i) is
asymptotically stable, while its linearization (ii) is Lyapunov-unstable (so, give an example how it can happen, and prove that it does happen in your
example). As another example, I claim that the combination (i) unstable, (ii)
asymptotically stable, is impossible (thus prove that asymptotical
stability of linearization implies asymptotical stability of the original