|Additions to this web site no longer count towards good deed points.
||Notes and Links
||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.
|| Monday: Separated equations, escape velocities. HW1. Tuesday: Escape velocities, changing source and target coordinates, homogeneous equations. Friday: Reverse engineering separated and exact equations.
|| Monday: Solving exact equations, integration factors. HW2. Tuesday: Statement of the Fundamental Theorem. Class Photo. Friday: Proof of the Fundamental Theorem.
|| 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.
||Monday is thanksgiving. Tuesday: Lagrange multiplyers and the isoperimetric inequality. HW4. Friday: More Lagrange multipliers, numerical methods.
|| 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.
|| Monday: Multiple roots, reduction of order, undetermined coefficients. Tuesday: From systems to matrix exponentiation. HW5. Term Test on Friday.
|| 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
|| Monday: Non-homogeneous systems. Tuesday: The Catalan numbers, power series, and ODEs. Friday: Global existence for linear ODEs, the Wronskian.
||Monday-Tuesday is UofT November break. HW7. Friday: Series solutions for .
|| Monday: 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..
|| 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.
|| Monday: More Sturm comparisons, changing the independent variable. Tuesday: Amplitudes of oscillations. Last class was on Tuesday!
||The Final Exam (time, place, style, office hours times)
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This assignment is due at the tutorial on Tuesday October 9. Here and everywhere, neatness counts!! You may be brilliant and you may mean just the right things, but if your readers have a hard time deciphering your work they will give up and assume it is wrong.
Task 0. Identify yourself in the Class Photo!
Task 1. Let be a sequence of functions defined on some set , and suppose that some sequence of non-negative reals is given such that for every , . Suppose also that is finite. Prove that the sequence is uniformly convergent.
Task 2. Find the extrema of the following functionals:
- subject to and .
- subject to and .
- subject to and .
- Postponed! subject to and and .
Task 3. A roach I once met was mortally afraid of walls, and so when it walked on my kitchen's floor, its speed was exactly proportional to its distance from the nearest wall (that is, very near a wall it crawled very slowly, while in the centre of the room it run around quickly and happily). As a step towards simplifying 's life, help it find the fastest path from one point in the upper half plane to another point in the upper half plane, assuming there is only one wall around, built along the -axis .
||Dror's notes above / Student's notes below
Solution to Task 1. --Twine 18:11, 24 October 2012 (EDT)
Let . We have that .
such that .
Choose such an N. Then .
Then , we have .
As this result is independent of our choice of x, we have by the cauchy criterion that is uniformly convergent and exists.
Solutions to HW3: Mathstudent