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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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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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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  .
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| 11
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Nov 19
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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..
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| 12
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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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Add your name / see who's in!
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Solving the complicated integral in the Brachistochroe integral
integral sqrt((d-y)/y) dy
= integral sqrt(d y-y^2)/y dy
For the integrand sqrt(d y-y^2)/y, complete the square:
= integral sqrt(d^2/4-(y-d/2)^2)/y dy
For the integrand sqrt(d^2/4-(y-d/2)^2)/y, substitute u = y-d/2 and du = dy:
= integral (2 sqrt(d^2/4-u^2))/(d+2 u) du
= 2 integral sqrt(d^2/4-u^2)/(d+2 u) du
For the integrand sqrt(d^2/4-u^2)/(d+2 u), (assuming all variables are positive) substitute u = 1/2 d sin(s) and du = 1/2 d cos(s) ds. Then sqrt(d^2/4-u^2) = sqrt(d^2/4-1/4 d^2 sin^2(s)) = 1/2 d cos(s) and s = sin^(-1)((2 u)/d):
= d^2/2 integral (cos^2(s))/(d sin(s)+d) ds
For the integrand (cos^2(s))/(d sin(s)+d), substitute p = tan(s/2) and dp = 1/2 sec^2(s/2) ds. Then transform the integrand using the substitutions sin(s) = (2 p)/(p^2+1), cos(s) = (1-p^2)/(p^2+1) and ds = (2 dp)/(p^2+1):
= d^2/2 integral (2 (1-p^2)^2)/((p^2+1)^3 ((2 d p)/(p^2+1)+d)) dp
Simplify the integrand (2 (1-p^2)^2)/((p^2+1)^3 ((2 d p)/(p^2+1)+d)) to get (2 (p-1)^2)/(d p^4+2 d p^2+d):
= d^2/2 integral (2 (p-1)^2)/(d p^4+2 d p^2+d) dp
= d^2 integral (p-1)^2/(d p^4+2 d p^2+d) dp
= d^2 integral (p-1)^2/(d (p^2+1)^2) dp
= d integral (p-1)^2/(p^2+1)^2 dp
For the integrand (p-1)^2/(p^2+1)^2, use partial fractions:
= d integral (1/(p^2+1)-(2 p)/(p^2+1)^2) dp
= d integral 1/(p^2+1) dp-2 d integral p/(p^2+1)^2 dp
For the integrand p/(p^2+1)^2, substitute w = p^2+1 and dw = 2 p dp:
= d integral 1/(p^2+1) dp-d integral 1/w^2 dw
The integral of 1/(p^2+1) is tan^(-1)(p):
= d tan^(-1)(p)-d integral 1/w^2 dw
= d tan^(-1)(p)+d/w+constant
Substitute back for w = p^2+1:
= (d ((p^2+1) tan^(-1)(p)+1))/(p^2+1)+C
Substitute back for p = tan(s/2):
= 1/2 d (cos(s)+2 tan^(-1)(tan(s/2))+1)+C
Substitute back for s = sin^(-1)((2 u)/d):
= 1/2 (sqrt(d^2-4 u^2)+2 d tan^(-1)((2 u)/(d (sqrt(1-(4 u^2)/d^2)+1)))+d)+C
Substitute back for u = y-d/2:
= d (-tan^(-1)((d-2 y)/(2 d sqrt((y (d-y))/d^2)+d)))+sqrt(y (d-y))+d/2+C
Factor the answer a different way:
= 1/2 (-2 d tan^(-1)((d-2 y)/(2 d sqrt((y (d-y))/d^2)+d))+2 sqrt(y (d-y))+d)+C
Which is equivalent for restricted y and d values to:
= y sqrt(d/y-1)-1/2 d tan^(-1)((sqrt(d/y-1) (d-2 y))/(2 (d-y)))+C
Syjytg 23:00, 11 September 2012 (EDT)
