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Week of...
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Notes and Links
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| 1
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Sep 10
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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.
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| 2
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Sep 17
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Monday: Separated equations, escape velocities. HW1. Tuesday: Escape velocities, changing source and target coordinates, homogeneous equations. Friday: Reverse engineering separated and exact equations.
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| 3
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Sep 24
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Monday: Solving exact equations, integration factors. HW2. Tuesday: Statement of the Fundamental Theorem. Class Photo. Friday: Proof of the Fundamental Theorem.
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| 4
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Oct 1
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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.
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| 5
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Oct 8
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Monday is thanksgiving. Tuesday: Lagrange multiplyers and the isoperimetric inequality. HW4. Friday: More Lagrange multipliers, numerical methods.
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| 6
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Oct 15
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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.
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| 7
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Oct 22
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Monday: Multiple roots, reduction of order, undetermined coefficients. Tuesday: From systems to matrix exponentiation. HW5. Term Test on Friday.
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| 8
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Oct 29
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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
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| 9
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Nov 5
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Monday: Non-homogeneous systems. Tuesday: The Catalan numbers, power series, and ODEs. Friday: Global existence for linear ODEs, the Wronskian.
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| 10
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Nov 12
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Monday-Tuesday is UofT November break. HW7. Friday: Series solutions for [math]\displaystyle{ y'=f(x,y) }[/math].
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| 11
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Nov 19
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Monday: [math]\displaystyle{ \pi }[/math] 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..
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Nov 26
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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.
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Dec 3
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Monday: More Sturm comparisons, changing the independent variable. Tuesday: Amplitudes of oscillations. Last class was on Tuesday!
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| F1
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Dec 10
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| F2
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Dec 17
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The Final Exam (time, place, style, office hours times)
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| Register of Good Deeds
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Add your name / see who's in!
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Disclamer: This is a student prepared note based on the lecure of Tuesday October 2nd.
For a function [math]\displaystyle{ y(x) }[/math] defined on [math]\displaystyle{ [a, b] }[/math] to be an extremum of [math]\displaystyle{ J(y) = \int_a^b F(x, y, y') dx }[/math], it must be that for any function [math]\displaystyle{ h(x) }[/math] defined on [math]\displaystyle{ [a, b] }[/math] that preserves the endpoints of [math]\displaystyle{ y }[/math] (that is, [math]\displaystyle{ h(a) = 0 }[/math] and [math]\displaystyle{ h(b) = 0 }[/math]), we have [math]\displaystyle{ \frac{d}{d \epsilon } J(y + \epsilon h) }[/math][math]\displaystyle{ |_{\epsilon = 0} = 0 }[/math].
[math]\displaystyle{ \frac{d}{d \epsilon } J(y + \epsilon h) }[/math][math]\displaystyle{ |_{\epsilon = 0} }[/math]
[math]\displaystyle{ = \frac{d}{d \epsilon } \int_a^b F(x, y + \epsilon h, y' + \epsilon h') dx |_{\epsilon = 0} }[/math]
Let [math]\displaystyle{ F_n }[/math] signify F differentiated with respect to its nth variable.
[math]\displaystyle{ = \int_a^b (F_1 \cdot 0 + F_2 \cdot h + F_3 \cdot h') dx |_{\epsilon = 0} }[/math]
[math]\displaystyle{ = \int_a^b (F_2(x, y, y') \cdot h + F_3(x, y, y') \cdot h') dx }[/math]
[math]\displaystyle{ = \int_a^b (F_2 \cdot h - [\frac{d}{dx} F_3] \cdot h) dx + F_3 \cdot h |_a^b }[/math] (integrating by parts)
Due to the constraints of [math]\displaystyle{ h(a) = 0 }[/math] and [math]\displaystyle{ h(b) = 0 }[/math], [math]\displaystyle{ F_3 \cdot h |_a^b = 0 }[/math].
[math]\displaystyle{ = \int_a^b (F_2 - \frac{d}{dx} F_3) \cdot h dx }[/math]
As this must be equal to 0 for all h satisfying the endpoint constraints, we must have that [math]\displaystyle{ F_2 = \frac{d}{dx} F_3 }[/math], or in other terms, [math]\displaystyle{ F_y - \frac{d}{dx} F_y' = 0 }[/math].
Special cases (without derivations):
In the case that F does not depend on y', we have [math]\displaystyle{ F_y = 0 }[/math]
In the case that F does not depend on y, we have [math]\displaystyle{ F_{y'} = c }[/math]
In the case that F does not depend on x, we have [math]\displaystyle{ F - y'F_{y'} = c }[/math]
