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Topics - Victor Ivrii

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436
MAT244 Announcements / Mid Term Coverage
« on: February 24, 2013, 04:39:41 PM »
Quote
What chapters are covered on the coming midterm?

As usual, i.e. all that was included in TT1 (both material in the textbook and handouts not marked by "for the curious student" including) plus what will be covered in the lectures during the coming week.

During the coming week we will start Chapter 7 at 7.5 and some of 7.4 . Note that the earlier parts of Chapter 7 belong to the prerequisite material on the linear algebra and is to be reviewed by students on their own. In the case of any problems I recommend to use the office hours during the coming week (during the coming week besides regular 12 hours we'll have 13 special office hours (by Graig Sinnamon).

Sincerely, -- Pierre Milman


437
MAT244 Announcements / Addition to Home Assignments
« on: February 24, 2013, 04:35:26 PM »
Section 4.4: added problems 5, 7 pages 244-245; due date Tuesday February 26.

438
Term Test 1 / Official solutions
« on: February 21, 2013, 04:47:31 PM »
There are "official" solutions

439
MAT244 Announcements / Retyping LN8
« on: February 21, 2013, 09:35:09 AM »
LN8 http://www.math.toronto.edu/courses/mat244h1/20131/MAT244-LN8.pdf is a scan (recently I OCRed it) of unknown text

We would like to have it retyped using LaTeX. Basic knowledge of LaTeX is definitely needed but nothing more as text will be edited anyway. It will be considered as a public service with karma points awarded. Those who have few (or none) of them are especially welcome.

440
MAT244 Announcements / TT1 marks have been posted
« on: February 20, 2013, 03:51:43 PM »
You may see your TT1 marks on BlackBoard

Immediately after Reading Week there will be extra office hours:
TT1 was graded by Craig Sinnamon, and he will have an extra office hours for you to get your test and discuss with him. Since MidTerm which costs twice as much as TT1 will be based on the same material + Chapter 7 you definitely should use this opportunity. Recall:

February 25, Mon, 18:30—20:30 Craig Sinnamon in BA6283
February 26, Tue, 15:00—16:0 Craig Sinnamon in BA6283 and 17:00—20:00 in BA6283
February 28, Thu, 18:30—20:30 Craig Sinnamon in BA6283
March 01,       Fri, 11:00—12:00 Craig Sinnamon in BA6283 and 16:00—20:00 in BA6283

441
Reading Week Challenge / Reading Week Bonus problem 4
« on: February 16, 2013, 10:40:07 AM »
Using Reading Week Bonus problem 3 find inequalities for two consecutive zeros $x_n$ and $x_{n+1}$ of Airy function satisfying equation
\begin{equation}
y''+xy=0,
\end{equation}
Then derive asymptotic formula for $x_n$ as $n\to +\infty$.

442
Reading Week Challenge / Reading Week Bonus problem 3
« on: February 16, 2013, 10:37:44 AM »
Using Reading Week Bonus problem 2 consider $Q$ as a constant which is the maximum (minimum) of $q(x)$ on interval $[x_n,x_{n+1}]$ where $x_n$ and $x_{n+1}$ are two consecutive zeros of $y(x)$ and estimate $(x_{n+1}-x_n)$ from above (from below--respectively).

443
Reading Week Challenge / Reading Week Bonus problem 2
« on: February 16, 2013, 10:35:08 AM »
Using Reading Week Bonus problem 1 prove (in its framework)  that if $y(x_0)=y(x_1)=0$ and $y(x)>0$ on $(x_0,x_1)$, $x_0<x_1$ then $z(x)$ has a $0$ somewhere on $(x_0,x_1)$ unless $z(x_0)=y(x_1)=0$ and $Q(x)=q(x)$ on $(x_0,x_1)$.

444
Reading Week Challenge / Reading Week Bonus problem 1
« on: February 16, 2013, 10:27:54 AM »
For "theoretical dudes" I suggest the sequence of the problems devoted comparison theorems and studying consecutive zeroes of  solutions to 2nd order linear homogeneous equations.

Consider two equations:
\begin{gather}
y'' + q(x)y=0,\\
z''+Q(x)z''=0
\end{gather}
with
\begin{equation}
Q(x)\ge q(x).
\end{equation}
Consider Wronskian $W(x):=W[y,z](x)$ and assuming that $y(x)>0$, $z(x)>0$ on interval $[a,b]$ derive differential inequality
\begin{equation}
W'  ?   0.
\end{equation}

445
MAT244 Announcements / No office hours during Reading Week
« on: February 15, 2013, 10:42:07 AM »
There will be no office hours during Reading Week (I am writing on behalf all instructors and TAs).

On Forum there will be Reading Week Challenge: posted Saturday February 16

Immediately after Reading Week there will be extra office hours:
TT1 will be graded by Craig Sinnamon, and he will have an extra office hours for you to get your test and discuss with him. Since MidTerm which costs twice as much as TT1 will be based on the same material + Chapter 7 you definitely should use this opportunity. Recall:

February 25, Mon, 18:30—20:30 Craig Sinnamon in BA6283
February 26, Tue, 15:00—16:0 Craig Sinnamon in BA6283 and 17:00—20:00 in BA6283
February 28, Thu, 18:30—20:30 Craig Sinnamon in BA6283
March 01,       Fri, 11:00—12:00 Craig Sinnamon in BA6283 and 16:00—20:00 in BA6283


446
Term Test 1 / MOVED: Advice on how to get faster?
« on: February 14, 2013, 04:43:16 PM »

447
Term Test 1 / TT1--Problem 4
« on: February 13, 2013, 10:41:28 PM »
Find  solution
\begin{equation*}
y^{(4)}+8y''+16y=0
\end{equation*}
satisfying initial conditions
\begin{equation*}
y(0)=1,\; y'(0)=y''(0)=y'''(0)=0.
\end{equation*}

448
Term Test 1 / TT1--Problem 3
« on: February 13, 2013, 10:39:58 PM »
Find the general solution for equation
\begin{equation*}
y'' + 4y'+5y =t e^{-2t}+ e^{-2t}\cos(t).
\end{equation*}

449
Term Test 1 / TT1--Problem 2
« on: February 13, 2013, 10:38:31 PM »
(a) Consider equation
\begin{equation*}
(\cos(t)+t\sin(t))y''-t\cos(t)y'+y\cos(t)=0.
\end{equation*}
Find wronskian $W=W[y_1,y_2](t)$ of two solutions such that $W(0)=1$.
 
(b) Check that  one of the solutions is $y_1(t)=t$. Find another solution $y_2$ such that $W[y_1,y_2](\pi/2)=\pi/2$
and $y_2(\pi/2)=0$.

450
Term Test 1 / TT1--Problem 1
« on: February 13, 2013, 10:36:58 PM »
 Find integrating factor and solve
\begin{equation*}
x\,dx +y (1+x^2+y^2)\,dy=0.
\end{equation*}

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