MATH 431-SUMMER 2012-COURSE LOG

Th 5/31  Overview of DE/ review of matrix algebra

F  6/1   Homogeneous linear systems: fundamental solutions/Normed vector spaces

M 6/4  Matrix exponential, fundamental matrices for diagonalizable systems

HW (1.4) 6, 8
HW (1.5) 5, 6

Theory (1.4) 2, 12, 14 (1.5) 9

Remark: Problems in the "theory" list are not to be turned in as homework. Think about them, and if you
believe you can do one of them, you may volunteer to present the solution in class (for extra HW credit).

Tu 6/5 Complex eigenvalues

HW(1.6) 7,8,9: matrices (c) and (d) only

W 6/6 Putzer algorithm

HW (1.7) 2d, 6, 8b

Theory (1.7) 9

Th 6/7 Non-homogeneous systems
HW (1.8) 4, 5
Theory (1.8) 3 (for problems 4,5)

F 6/8 Stability, coupled oscillators--HW 1 due

HW (1.9) 2
Theory (1.9) 8, 11 (find a 3X3 example with real eigenvalues), 12

M 6/11 Phase diagrams for 2D linear systems

HW (2.3) 9, (2.4) 5,6

Tu 6/12 Periodic coefficients (1.10)

HW (1.10) 9,10,11
Theory: 6,7,8 (will be discussed on 6/13)

W  6/13 hyperbolic systems, stable/unstable spaces; coupled oscillations

Th 6/14 HW2 due/ discussion of HW problems/ outline of qualitative analysis in 2D/ Matlab graphics
(go to Professor John Polking's page at math.rice.edu and download dfield8 and pplane8 for Matlab).

F 6/15 Exam 1
Exam 1 solutions

M 6/18  Autonomous systems: critical points, linearization (Hartman's theorem), global qualitative analysis
(basins of attraction, saddle separatrices--asymptotic behavior, interval of existence)
Problems for chapter 2(HW3)

Tu 6/19
Systems with periodic orbits/ Evolution of functions along solutions: conserved quantities, Liapunov functions
HW3: (2.6) 1,2
Theory: (2.6) 3,4,5

W 6/20
Proper Liapunov functions/ Gradient systems/ Lotka-Volterra (2.8), Bendixson-Dulac criterion
HW3: two problems have been added to the problem set linked to the 6/18 lecture
Theory: show that a gradient system cannot have non-constant periodic solutions.

Th 6/21
Mechanics/ Poincare-Bendixson theorem and applications; Lienard's theorem (statement)
HW3: (2.7)8, plus one problem added to the problem set linked to 6/18
Theory: (2.7)10

F 6/22 Examples: Lienard's theorem, predator-prey, proper Liapunov functions, gradient systems, Hopf bifurcation
HW 3 due

M 6/25 Linear boundary-value problems on an interval/ Green's functions/ solution of the non-homog. BVP.
HW4  I: (4.2)  1a) d) e): do it either by the Green's function method or directly, but compute the Green's function anyway
(with zero boundary values).
For extra credit (0.5 pts.): include a plot of the Green's function G(t,tau), for a chosen value of tau.

HW4 II: Suppose the linear operator L[y]=y''+a(t)y'+b(t)y does not have a first-order term: a(t)=0. Show that, in this case,
we have the symmetry property: G(t,tau)=G(tau,t), for all t,tau in (t_0,t_1). (Hint: use the explicit expression for G(t,tau), and
the DE satisfied by the Wronskian.)

Tu 6/26 Sturm-Liouville eigenvalue problems: orthogonality of the eigenfunctions/ Sturm's oscillation and comparison theorems
HW4 III: (4.3) 1, (4.5) 5, (4.6) 4

W 6/27 Existence of infinitely many eigenvalues for S-L problems/ transformations
Example: non-integrable positive potentials yield oscillatory solutions

HW 4 IV: (i) Show that every solution of the Airy equation u''+xu=0, u=u(x), x in R, has an infinite number of zeros for x>0
and at most one for x<0.
(ii) Show the Sturm-Liouville problem defined by Airy's equation on the interval [0,1] (with zero boundary conditions) does not
have negative eigenvalues.

Th 6/28: Existence theory, preliminaries: fixed points for contractions (3.2, 3.3)
Practice problems (not due as HW): (3.3) 2,3,4,5

F 6/29: Existence/uniqueness for the scalar IVP by Picard iteration/Gronwall's inequality/continuous dependence (3.4)
Practice problems (not due as HW): (3.4) 1,2
HW 4 due

M  7/2: Examples, discussion of problems

Tu 7/3 FINAL EXAM (Chapters 2 (from 2.5), 3, 4)
Exam 2
solutions