Lectures:

Solve: Problems 4, 5 of Section 1.15, and Problem 2 of Section 1.18.

Solve: Problem 1 of Section 2.6, Problem 1 of Section 2.7, and Problem 1 of Section 3.2.

(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.

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.

(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,

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

(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

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.

(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

(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.

(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.

(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