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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 lecures of Friday, September 28th and Monday October 1st.
Lipschitz
Def. $f:\mathbb {R} _{y}\rightarrow \mathbb {R}$ is called Lipschitz if $\exists \epsilon >0,k>0$ (a Lipschitz constant of f) such that $y_{1}y_{2}<\epsilon \implies f(y_{1})f(y_{2})\leq ky_{1}y_{2}$.
Note that any function that is Lipschitz is uniformly continuous, and that if a function f and its derivative are both continous on a compact set then f is Lipschitz.
Statement of Existence and Uniqueness Theorem
Thm. Existence and Uniqueness Theorem for ODEs
Let $f:\mathbb {R} =[x_{0}a,x_{0}+a]\times [y_{0}b,y_{0}+b]\rightarrow \mathbb {R}$ be continuous and uniformly Lipschitz relative to y. Then the equation $\Phi '=f(x,\Phi )$ with $\Phi (x_{0})=y_{0}$ has a unique solution $\Phi :[x_{0}\delta ,x_{0}+\delta ]\rightarrow \mathbb {R}$ where $\delta =min(a,^{b}/_{M})$ where M is a bound of f on $\mathbb {R}$.
Proof of Existence
This is proven by showing the equation $\Phi (x)=y_{0}+\int _{x_{0}}^{x}f(t,\Phi (t))dt$ exists, given the noted assumptions.
Let $\Phi _{0}(x)=y_{0}$ and let $\Phi _{n}(x)=y_{0}+\int _{x_{0}}^{x}f(t,\Phi _{n1}(t))dt$. IF we can prove the following three claims, we have proven the theorem. The proofs of these claims will follow below.
Claim 1: $\Phi _{n}$ is welldefined. More precisely, $\Phi _{n}$ is continuous and $\forall x\in [x_{0}\delta ,x_{0}+\delta ]$, $\Phi _{n}(x)y_{0}\leq b$ where b is as referred to above.
Claim 2: For $n\geq 1$, $\Phi _{n}(x)\Phi _{n1}(x)\leq {\frac {Mk^{n1}}{n!}}xx_{0}^{n}$.
Claim 3: if $\Phi _{n}(x)$ is a series of functions such that $\Phi _{n}(x)\Phi _{n1}(x)<c_{n}$, with $\sum _{n=1}^{\infty }c_{n}$ equal to some finite number, then $\Phi _{n}$ converges uniformly to some function $\Phi$
Using these three claims, we have shown that the solution $\Phi (x)$ exists.
Proofs of Claims
Proof of Claim 1:
The statement is trivially true for $\Phi _{0}$. Assume the claim is true for $\Phi _{n1}$. $\Phi _{n}$ is continuous, being the integral of a continuous function.
$\Phi _{n}y_{0}$
$=\int _{x_{0}}^{x}f(t,\Phi _{n1}(t))dt$
$\leq \int _{x_{0}}^{x}f(t,\Phi _{n1}(t))dt$
$\leq \int _{x_{0}}^{x}Mdt=Mx_{0}x$
$\leq M\delta$
$\leq M\cdot {\frac {b}{M}}$
$=b$
$\Box$
Proof of Claim 2:
$\Phi _{n}(x)\Phi _{n1}(x)$
$=\int _{x_{0}}^{x}f(t,\Phi _{n1}(t))dt\int _{x_{0}}^{x}f(t,\Phi _{n2}(t))dt$
$\leq \int _{x_{0}}^{x}f(t,\Phi _{n1}(t))f(t,\Phi _{n2}(t))dt$
$\leq \int _{x_{0}}^{x}k\Phi _{n1}(t)\Phi _{n2}(t)dt$
$\leq \int _{x_{0}}^{x}k{\frac {Mk^{n2}}{(n1)!}}tx_{0}^{n1}dt$
$={\frac {Mk^{n1}}{(n1)!}}\int _{0}^{xx_{0}}t^{n1}dt$
$={\frac {Mk^{n1}}{n!}}xx_{0}^{n}$
$\Box$
Note that the sequence $c_{n}={\frac {Mk^{n1}}{n!}}xx_{0}^{n}$ has $\sum _{n=1}^{\infty }c_{n}$ equal to some finite number.
Proof of Claim 3: Assigned in Homework 3, Task 1, see page for solutions.
Proof of Uniqueness
Suppose $\Phi$ and $\Psi$ are both solutions. Let $\mathrm {X} (x)=\Phi (x)\Psi (x)$.
$\mathrm {X} (x)=\Phi (x)\Psi (x)=\int _{x_{0}}^{x}(f(x,\Phi (x))f(x,\Psi (x)))dx\leq \int _{x_{0}}^{x}k\Phi (x)\Psi (x)dx$
We have that $\mathrm {X} \leq k\int _{x_{0}}^{x}\mathrm {X} (x)dx$ for some constant k, which means $\mathrm {X} '\leq k\mathrm {X}$, and that $\mathrm {X} (x)\geq 0$.
Let $U(x)=e^{kx}\int _{x_{0}}^{x}\mathrm {X} (x)dx$. Note that $U(x_{0})=0$ as in this case we are integrating over an empty set, and that U thus defined has $U(x)\geq 0$. Then
$U'(x)=ke^{kx}\int _{x_{0}}^{x}\mathrm {X} (x)dx+e^{kx}\mathrm {X} (x)=e^{kx}(\mathrm {X} (x)k\int _{x_{0}}^{x}\mathrm {X} (x)dx)\leq 0$
Then $U(x_{0})=0\land U'(x)=0\implies U(x)\leq 0$, and $0\leq U(x)\leq 0\implies U(x)\equiv 0\implies \mathrm {X} (x)\equiv 0\implies \Phi (x)\equiv \Psi (x)$.
$\Box$