12-267/Homework Assignment 6

DBN: Classes: 2012-13: 12-267 / Navigation

#	Week of...	Notes and Links
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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 Euler-Lagrange. Friday: Reductions of Euler-Lagrange.
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, Runge-Kutta, 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: Non-homogeneous systems. Tuesday: The Catalan numbers, power series, and ODEs. Friday: Global existence for linear ODEs, the Wronskian.
10	Nov 12	Monday-Tuesday 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: Non-oscillation, 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 in class on Friday November 9. Here and everywhere, neatness counts!! You may be brilliant and you may mean just the right things, but if your readers have a hard time deciphering your work they will give up and assume it is wrong.

Task 1. Draw the phase portraits for the following systems, near $(x,y)=(0,0)$ :

${\begin{cases}{\dot {x}}=2x+y\\{\dot {y}}=-x+4y\end{cases}}$ .
${\begin{cases}{\dot {x}}=4x-5y\\{\dot {y}}=4x-4y\end{cases}}$ .
${\begin{cases}{\dot {x}}=x-2y\\{\dot {y}}=-2x+4y\end{cases}}$ .
${\begin{cases}{\dot {x}}=-x+y\\{\dot {y}}=-5x+3y\end{cases}}$ .
${\begin{cases}{\dot {x}}=-5x+4y\\{\dot {y}}=-8x+7y\end{cases}}$ .

Task 2. Draw the phase portrait of the system

${\begin{cases}{\dot {x}}=17+x-9y+\sin(2-2x-y+xy)\\{\dot {y}}=7+2x-5y+\cos(x-1)\end{cases}}$

near the point $(x,y)=(1,2)$ .

Task 3. Solve using diagonalization (one solution is enough):

$v'={\begin{pmatrix}2&-1\\3&-2\end{pmatrix}}v+{\begin{pmatrix}e^{t}\\t\end{pmatrix}}$ .
$v'={\begin{pmatrix}2&-5\\1&-2\end{pmatrix}}v+{\begin{pmatrix}-\cos t\\\sin t\end{pmatrix}}$ .