© | Dror Bar-Natan: Classes: 2012-13: Math 267 ODEs: < >

# 121102 Video

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Phase Portraits.

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# Week of... Notes and Links Additions to this web site no longer count towards good deed points. 1 Sep 10 About This Class. : Introduction and the Brachistochrone. : More on the Brachistochrone, administrative issues. Tuesday Notes. : Some basic techniques: first order linear equations. 2 Sep 17 : Separated equations, escape velocities. HW1. : Escape velocities, changing source and target coordinates, homogeneous equations. : Reverse engineering separated and exact equations. 3 Sep 24 : Solving exact equations, integration factors. HW2. : Statement of the Fundamental Theorem. Class Photo. : Proof of the Fundamental Theorem. 4 Oct 1 : Last notes on the fundamental theorem. HW3. : The chain law, examples of variational problems. : Deriving Euler-Lagrange. : Reductions of Euler-Lagrange. 5 Oct 8 Monday is thanksgiving. : Lagrange multiplyers and the isoperimetric inequality. HW4. : More Lagrange multipliers, numerical methods. 6 Oct 15 : Euler and improved Euler. : Evaluating the local error, Runge-Kutta, and a comparison of methods. : Numerical integration, high order constant coefficient homogeneous linear ODEs. 7 Oct 22 : Multiple roots, reduction of order, undetermined coefficients. : From systems to matrix exponentiation. HW5. Term Test on Friday. 8 Oct 29 : The basic properties of matrix exponentiation. : Matrix exponentiation: examples. : Phase Portraits. HW6. Nov 4 was the last day to drop this class 9 Nov 5 : Non-homogeneous systems. : The Catalan numbers, power series, and ODEs. : Global existence for linear ODEs, the Wronskian. 10 Nov 12 Monday-Tuesday is UofT November break. HW7. : Series solutions for $y'=f(x,y)$. 11 Nov 19 : $\pi$ is irrational, more on the radius of convergence. : Airy's equation, Fuchs' theorem. : Regular singular points. HW8. : Discussion of regular singular points.. 12 Nov 26 : Frobenius series by computer. Qualitative Analysis Handout (PDF). : The basic oscillation theorem. Handout on the Frobenius Method. HW9. : Non-oscillation, Sturm comparison. 13 Dec 3 : More Sturm comparisons, changing the independent variable. : Amplitudes of oscillations. Last class was on Tuesday! F1 Dec 10 F2 Dec 17 The Final Exam (time, place, style, office hours times) Register of Good Deeds Add your name / see who's in!
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0:00:00  Handwritten lecture notes by Ktnd3: 12-267(lecture22).PDF
0:00:52  The front side of the handout is at PhasePortraits.pdf.
0:00:54  The back side of the handout is at QuadraticPortraits.html.
0:02:53  Motivation for studying systems of linear equations: most things are linear on small scales
0:08:13  Taylor expansion of a vector field around an equilibrium point
0:12:44  The behavior of a linear system depends on the eigenvalues of the matrix
0:18:48  Determining the trajectories of different points
0:21:49  A phase portrait is to differential equations as the graph of a function is to calculus
0:32:39  Repeated eigenvalue with two linearly independent eigenvectors
0:37:03  Eigenvalues are complex conjugates
0:45:00  The case when we can't diagonalize A (Jordan canonical form)