Fall-11. Math 123 (ccn 54356): Ordinary Differential Equations

Instructor: Alexander Givental
Lectures:
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.

HOMEWORK

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 Section 3.2.
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, where k=0,1,2,3,....
(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 the equaionn
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 its linearization. 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 equilibrium).