12-267/The Final Exam

DBN: Classes: 2012-13: 12-267 / Navigation Additions to this web site no longer count towards good deed points. # Week of... Notes and Links 1 Sep 10 About This Class.  Monday: Introduction and the Brachistochrone.  Tuesday: More on the Brachistochrone, administrative issues. Tuesday Notes.  Friday: Some basic techniques: first order linear equations. 2 Sep 17  Monday: Separated equations, escape velocities. HW1.  Tuesday: Escape velocities, changing source and target coordinates, homogeneous equations.  Friday: Reverse engineering separated and exact equations. 3 Sep 24  Monday: Solving exact equations, integration factors. HW2.  Tuesday: Statement of the Fundamental Theorem. Class Photo.  Friday: Proof of the Fundamental Theorem. 4 Oct 1  Monday: Last notes on the fundamental theorem. HW3.  Tuesday Hour 1: The chain law, examples of variational problems.  Tuesday Hour 2: Deriving Euler-Lagrange.  Friday: Reductions of Euler-Lagrange. 5 Oct 8 Monday is thanksgiving.  Tuesday: Lagrange multiplyers and the isoperimetric inequality. HW4.  Friday: More Lagrange multipliers, numerical methods. 6 Oct 15  Monday: Euler and improved Euler.  Tuesday: Evaluating the local error, Runge-Kutta, and a comparison of methods.  Friday: Numerical integration, high order constant coefficient homogeneous linear ODEs. 7 Oct 22  Monday: Multiple roots, reduction of order, undetermined coefficients.  Tuesday: From systems to matrix exponentiation. HW5. Term Test on Friday. 8 Oct 29  Monday: The basic properties of matrix exponentiation.  Tuesday: Matrix exponentiation: examples.  Friday: Phase Portraits. HW6. Nov 4 was the last day to drop this class 9 Nov 5  Monday: Non-homogeneous systems.  Tuesday: The Catalan numbers, power series, and ODEs.  Friday: Global existence for linear ODEs, the Wronskian. 10 Nov 12 Monday-Tuesday is UofT November break. HW7.  Friday: Series solutions for $y'=f(x,y)$. 11 Nov 19  Monday: $\pi$ is irrational, more on the radius of convergence.  Tuesday (class): Airy's equation, Fuchs' theorem.  Tuesday (tutorial): Regular singular points. HW8.  Friday: Discussion of regular singular points.. 12 Nov 26  Monday: Frobenius series by computer. Qualitative Analysis Handout (PDF).  Tuesday: The basic oscillation theorem. Handout on the Frobenius Method. HW9.  Friday: Non-oscillation, Sturm comparison. 13 Dec 3  Monday: More Sturm comparisons, changing the independent variable.  Tuesday: 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!

Our final exam is coming up. It will take place on Tuesday December 18th, from 7PM until 10PM (believe me that wasn't my decision), at East Hall, University College, 15 King's College Circle.

Sample Final

See SampleFinal.pdf.

Content and Style

It will consist of 5-6 questions (each may have several parts) on everything that we have covered in class this semester:

• Basic techniques: linear, separated, exact equations, integrating factors, etc.
• The Fundamental Theorem (existence and uniqueness).
• Calculus of Variations.
• Numerical methods.
• High order linear equations with constant coefficients, "undetermined coefficients".
• Systems of linear equations with constant coefficients, matrix exponentiation, phase portraits, Wronskians, non-homogeneous systems.
• Series solutions and regular singular points.
• Qualitative analysis.

As for the style -

• You can expect to be asked to reproduce some proofs that were given in class.
• You can expect some fresh things to prove, though generally not as hard as the previous type of proofs.
• You can expect questions (or parts of questions) that will be identical or nearly identical to questions that were assigned for homework.
• You can expect some calculations (but nothing that will require a calculator).

Basic calculators (not capable of displaying text or sounding speech) will be allowed but will not be necessary. You may wish to bring one nevertheless, as under pressure $5+7$ often comes out to be $13$.

Remember. Neatness counts! Organization counts! Language counts! Proofs are best given as short and readable essays; without the English between the formulas one never knows how to interpret those formulas. When you write, say, "$x\in [0,1]$", does it mean "choose $x\in [0,1]$", or "we've just proven that $x\in [0,1]$", or "assume by contradiction that $x\in [0,1]$", or "for every $x\in [0,1]$" or "there exists $x\in [0,1]$"? If you don't say, your reader has no way of knowing. Also remember that long and roundabout solutions of simple problems, full of detours and irrelevant facts, are often an indication that their author didn't quite get the point, even if they are entirely correct. Avoid those!

Office Hours

Jordan and I will hold pre-exam office hours as follows:

• Friday December 14, 3PM-5:30PM, with Jordan at the Math Lounge (the round room on the 6th floor of Bahen).
• Monday December 17, 10:30AM-11:30AM, with Dror at Bahen 6178.
• Monday December 17, 3PM-5:30PM, with Jordan at the Math Lounge.
• Tuesday December 18, 10AM-Noon and 3PM-5PM, with Dror at Bahen 6178.