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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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This assignment is due at the tutorial on Tuesday October 2. Here and everywhere, neatness counts!! You may be brilliant and you may mean just the right things, but if the your readers will be having hard time deciphering your work they will give up and assume it is wrong.
Task 0. Identify yourself in the Class Photo!
Task 1. Solve the following differential equations:
 $x^{2}y^{3}+x(1+y^{2})y'=0$ (hint: try $\mu =x^{\alpha }y^{\beta }$).
 $dx+({\frac {x}{y}}\sin y)dy=0$.
 $(x^{2}+3xy+y^{2})dxx^{2}dy=0$.
 ${\frac {dy}{dx}}={\frac {2yx+5}{2xy4}}$ (hint: consider trying $x_{1}=x+\alpha$ and $y_{1}=y+\beta$ for good $\alpha$, $\beta$).
 $y'={\frac {y^{3}}{12xy^{2}}}$ with $y(0)=1$.
 ${\frac {dy}{dx}}={\frac {2y+{\sqrt {x^{2}y^{2}}}}{2x}}$.
Task 2. Let $M$ and $N$ be differentiable functions of $x$ and $y$.
 Show that if ${\frac {N_{x}M_{y}}{xMyN}}$ depends only on $xy$, then the differential equation $Mdx+Ndy=0$ has an integrating factor of the form $\mu (xy)$, where $\mu$ is a function of a single variable.
 Find a condition on $M$ and $N$ that would imply that the differential equation $Mdx+Ndy=0$ would have an integrating factor of the form $\mu (x+y)$, where $\mu$ is a function of a single variable.
Task 3. The equation $y'+p(x)y=q(x)y^{n}$ is called a "Bernoulli Equation".
 Explain why you already know how to solve the Bernoulli equation when $n=0$ and when $n=1$.
 Show that if $n\neq 0,1$, then the substitution $v=y^{1n}$ reduces the Bernoulli equation to an equation you already know how to solve.
 Solve the equation $x^{2}y'+2xyy^{3}=0$ (for $x>0$).
Task 4. Find an example of a nondifferentiable function which is nevertheless Lipschitz.

Dror's notes above / Student's notes below


Solution to Task 2 Twine 18:44, 24 October 2012 (EDT)
Part 1
What we want is an integration factor $\mu$ such that $(\mu M)_{y}=(\mu N)_{x}$. Let $xy=z$
When $\mu$ is a function of z, ${\frac {dv}{dx}}={\frac {dv}{dz}}{\frac {dz}{dx}}={\frac {dv}{dz}}y$.
Similarly ${\frac {dv}{dy}}={\frac {dv}{dz}}x$.
Then we have
$(\mu M)_{y}=(\mu N)_{x}$
$\iff \mu _{z}xM\mu _{xy}yN=\mu (N_{x}M_{y})$
$\iff {\frac {\mu _{z}}{mu}}={\frac {N_{x}M_{y}}{xMyN}}$
If the right hand side depends only on xy, we can get
$\mu (xy)=e^{\int {\frac {N_{x}M_{y}}{xMyN}}d(xy)}$
which satisfies the requirements of an integrating factor.
Part 2
As in Part 1, we need $(\mu M)_{y}=(\mu N)_{x}$. Let $z=x+y$
When $\mu$ is a function of $x+y$, ${\frac {d\mu }{dx}}={\frac {d\mu }{dz}}{\frac {dz}{dx}}={\frac {d\mu }{dz}}$.
Similarly ${\frac {d\mu }{dy}}={\frac {d\mu }{dz}}$
Then we have
$(\mu M)_{y}=(\mu N)_{x}$
$\iff {\frac {\mu _{z}}{\mu }}={\frac {N_{x}M_{y}}{MN}}$
If the right hand side of this equation depends only on z (that is, only on (x+y)), then we have
$\mu (x+y)=e^{\int {\frac {N_{x}M_{y}}{MN}}d(x+y)}$
which satisfies the requirements of an integrating factor.
Solution to HW2: Mathstudent