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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 EulerLagrange. Friday: Reductions of EulerLagrange.

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, RungeKutta, 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: Nonhomogeneous systems. Tuesday: The Catalan numbers, power series, and ODEs. Friday: Global existence for linear ODEs, the Wronskian.

10

Nov 12

MondayTuesday 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: Nonoscillation, 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)

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Disclamer: This is a student prepared note based on the lecure of Tuesday October 2nd.
For a function $y(x)$ defined on $[a,b]$ to be an extremum of $J(y)=\int _{a}^{b}F(x,y,y')dx$, it must be that for any function $h(x)$ defined on $[a,b]$ that preserves the endpoints of $y$ (that is, $h(a)=0$ and $h(b)=0$), we have ${\frac {d}{d\epsilon }}J(y+\epsilon h)$$_{\epsilon =0}=0$.
${\frac {d}{d\epsilon }}J(y+\epsilon h)$$_{\epsilon =0}$
$={\frac {d}{d\epsilon }}\int _{a}^{b}F(x,y+\epsilon h,y'+\epsilon h')dx_{\epsilon =0}$
Let $F_{n}$ signify F differentiated with respect to its nth variable.
$=\int _{a}^{b}(F_{1}\cdot 0+F_{2}\cdot h+F_{3}\cdot h')dx_{\epsilon =0}$
$=\int _{a}^{b}(F_{2}(x,y,y')\cdot h+F_{3}(x,y,y')\cdot h')dx$
$=\int _{a}^{b}(F_{2}\cdot h[{\frac {d}{dx}}F_{3}]\cdot h)dx+F_{3}\cdot h_{a}^{b}$ (integrating by parts)
Due to the constraints of $h(a)=0$ and $h(b)=0$, $F_{3}\cdot h_{a}^{b}=0$.
$=\int _{a}^{b}(F_{2}{\frac {d}{dx}}F_{3})\cdot hdx$
As this must be equal to 0 for all h satisfying the endpoint constraints, we must have that $F_{2}={\frac {d}{dx}}F_{3}$, or in other terms, $F_{y}{\frac {d}{dx}}F_{y}'=0$.
Special cases (without derivations):
In the case that F does not depend on y', we have $F_{y}=0$
In the case that F does not depend on y, we have $F_{y'}=c$
In the case that F does not depend on x, we have $Fy'F_{y'}=c$