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Topics - Victor Ivrii

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1
Quiz-1 / MOVED: MAT24f4 TUT5103 Quiz2
« on: October 07, 2019, 05:32:24 AM »

2
Quiz-1 / MOVED: MAT24f4 TUT5103 Quiz2
« on: October 07, 2019, 05:32:12 AM »

3
MAT244--Lectures & Home Assignments / Existence and Uniqueness Theorem
« on: September 24, 2019, 10:56:17 AM »
Answering question on discussions

Definitely the role of the Existence and Uniqueness Theorems are much more important than Limits in Manual Computations (and honestly, I have no idea what "Manual Computation" means). However this role is more theoretical both in ODEs and PDEs. Still, if we are talking about numerical solutions (taught in different classes, we skip Chapter 8, and briefly look at section 2.7) we need to be pretty sure that the object we are trying to find exist and we find all of them.

For centuries from I. Newton (who introduced ODEs) mathematicians did not care much about existence, because they were looking for solutions of real life problems and believed in existence and also because the rigorous apparatus of Real Analysis which allows to prove such theorems came into existence only in 19th century. You may want to look at very sketchy Lecture_Note_to_Section_2.8_Existence-Uniqueness_Theorem.

Uniqueness is a different matter: mathematicians observed that the solution to the Cauchy problem is not necessarily unique (remember, that the general solution to the 1st order ODE is $x=\varphi(t;C)$ or $\Phi(x, t; C)=0$ in the explicit and implicit form correspondingly and we need to specify one solution one needs to impose an extra condition; f.e. $x(t_0)=x_0$. They discovered that there could be a singular solution which is not a regular solution which means that it cannot be obtained from the general solution by freezing $C$ but which in each point coincides with some (depending on the point) regular solution. In more details see Lecture_Note_to_Chapter_2_Singular_Solutions.

Both of these lecture notes are optional

4
In two examples were made errors.

A. Consider equation
\begin{equation}
y'= -\frac{y}{x}+y^2.
\label{1}
\end{equation}
It is Bernoulli equation. We solve it by the method of variation of the constant. Consider first corresponding linear homogeneous equation
\begin{equation}
y'= -\frac{y}{x}.
\label{2}
\end{equation}
It has a solution
\begin{equation}
y= Cx^{-1}
\label{3}
\end{equation}
with constant $C$ (do it by yourself!). Now consider (\ref{3}) with $C$ which is not a constant (variation!) and plug it into (\ref{1}).
\begin{gather*}
(Cx^{-1})' =-x^{-1}(Cx^{-1})+(Cx^{-1})^2\implies C'x^{-1}-Cx^{-2}=-Cx^{-2} + C^2x^{-2}\implies C' =C^2x^{-1}\\
\implies  \frac{dC}{C^2}=\frac{dx}{x}\implies -\frac{1}{C}=\ln(x)+c
\end{gather*}
where $c=\mathsf{const}$. Then  $C=-\dfrac{1}{\ln (x)+c}$ and plugging into (\ref{3}) we get
\begin{equation*}
\boxed{ y=-\frac{1}{x(\ln (x)+c)}.}
\end{equation*}

B. Consider equation
\begin{equation}
y'- \tan(x) y =\cos(x).
\label{4}
\end{equation}
We solve it by the method of integrating factor. Multiplying (\ref{4}) by unknown yet factor $\mu=\mu(x)$ we get
\begin{equation}
\mu y' - \mu \tan (x) y = \mu \cos(x).
\label{5}
\end{equation}
We want the left hand expression to be $\mu y'+\mu 'y$, which means
\begin{equation*}
\mu'=-\mu \tan(x) \implies \frac{d\mu}{\mu}= \tan(x)\,dx \implies \ln (\mu) =-\int \tan(x)\,dx = -\int\frac{\sin(x)}{\cos(x)}\,dx = \ln (\cos(x)).
\end{equation*}
You must know this integral. We do not need any constant since we need just one integrating factor.

So $\mu =\cos(x)$. (\ref{5}) is now
\begin{gather*}
\cos(x)y'-\sin (x)y =\cos^2(x)\implies \bigl(\cos(x)y\bigr)'= \cos^2(x)\implies
\cos(x)y = \int \cos^2(x)\,dx = \int \frac{1+\cos(2x)}{2}\,dx =\\
 \frac{x}{2}+\frac{\sin (2x)}{4}+C=
\frac{x}{2}+\frac{\sin (x)\cos(x)}{2}+C.
\end{gather*}
You must know simple trigonometric formulae. Then
\begin{equation*}
\boxed{y= \frac{1}{2}(x+2C)\sec(x)+ \frac{1}{2}\sin (x).}
\end{equation*}

5
APM346--Misc / Welcome to APM346
« on: January 07, 2019, 06:30:22 AM »
Welcome to this forum!

The main purpose of this forum is not the communication with instructors but peer-to-peer. I noticed that some of you answer the questions of your peers and I really appreciate, but I would like to see more -- much more -- of this!

It runs as aid to the class website. More permanent and systematic information is placed on website, while forum is for discussion. Please read before registering!

Registration
  • You can select any username (login) you want (take a rather short one). Username is what only you (and Admin) can see.
  • However  change your (screen) Name  (the one which everybody sees) to one easily identifiable (Admin should be able to identify you with a student enrolled to this course--and matching BlackBoard name; otherwise your posting privileges could be reduced or revoked):  Open Profile > Account Settings and look for "Name".
  • It would be preferable to use your University of Toronto email address.
  • BTW you can have a cool avatar too. Please do not display any confidential info.
  • You cannot change the loginname (used for login).  But there is no need.  Nobody except you knows your password - even Admins - but Admins don't need  to know it.

Some forum rules
  • Please use different boards according to their description (I will create new ones if needed).
  • Do not hijack topics: if a topic is devoted to some question, do not post anything which is not related. Start a new topic.
  • On the other hand, do not start new topic answering to the existing post; use the same topic.
  • Do not put in the same post several not related things - make several posts instead (in different topics).
  • Do not post solution which coincides with the solution already posted by someone else; exception: typed solution beats the scanned one.
  • Do not spam (post irrelevant messages, especially to promote something), flood (post many messages with very little or no sense, like "me too", "I like this post", ...) or flame (messages, containing personal attacks and heated arguments).
  • Do not thank instructors.
  • Do not post snapshots of paper documents, scan them (with the scanner or smartphone).
  • Do not use red color (it belongs to Admins and Moderators), large fonts
  • Read and search before postings.
  • Use common sense.

Karma
  • This semester karma does not translates into Bonus mark points (there are plenty reasons for this) 
  • On the other hand, your posts and active participation in the class will be a decisive factor if you ask me for a recommendation letter for a Graduate School. To write such letter I need to know something about you beyond your marks (and your final mark is in the transcript anyway). Participation in the forum automatically leaves a trace.
     

Have questions?
  • If you have a question which you think is of general interest--ask it on the Forum rather than via email.

MathJax
  • Both this forum and website use MathJax to display mathematics.

6
Technical Questions / emoji
« on: December 20, 2018, 08:37:39 PM »
😈 👿 👹 👺 💀 👻 👽 🤖 💩 😺 😸 😹 😻 😼 😽 🙀 😿 😾

7
Final Exam / FE-P6
« on: December 18, 2018, 06:22:02 AM »
Calculate for real $n>1$
$$
I:= \int_0^\infty\frac{dx}{1+x^n}.
$$

Hint:  Consider
$$
\int_\gamma \frac{dz}{1+z^n}
$$
with with an arc of radius $R\to \infty$ and an angle $\alpha=\frac{2\pi}{n}$. Express the integral over the second straight segment through integral over the first one.

8
Final Exam / FE-P5
« on: December 18, 2018, 06:18:35 AM »
Consider $P(z)= z^3 +2z -3-i$ and, using the argument theorem and Rouché's theorem calculate the number of its roots in each of the following domains:

(a)  $\{z\colon |z-1|<1\}$;

(b)  $\{z\colon |z-1|>1, |z|<2\}$,

(c) $\{z\colon |z|>2\}$.

9
Final Exam / FE-P4
« on: December 18, 2018, 06:17:32 AM »
(a) Find the Möbius transformation (fractional-linear transformation) $f(z)$ mapping the unit disk $\{z\colon |z|<1\}$ onto itself, such that $f(0)=\frac{1}{2}$ and $f(1)=-1$.

(b) Find the fixed points of $f$ (points s.t. $f(z)=z$)

(c) Find the stretch ($|f'(z)|$) and the rotation angle ($\arg(f'(z))$) of $f$ at $z$.

10
Final Exam / FE-P3
« on: December 18, 2018, 06:14:31 AM »
Find all singular points, classify them, and find residues at these points of
$$
f(z)= \tan (z) + z\cot^2(z);
$$
infinity included.

11
Final Exam / FE-P2
« on: December 18, 2018, 06:13:19 AM »
(a) Check that circles $\{z\colon |z|=r\}$ (with $0<r<1$) are mapped onto confocal ellipses
$\{w=u+iv\colon \frac{u^2}{a^2}+\frac{v^2}{b^2}=1\}$ with $a^2-b^2=1$ and find $a=a(r)$ and $b=b(r)$.

(b) Check that segments $\{z\colon z= e^{i\theta}r,\ r\in (-1,1)\}$  are mapped onto confocal hyperbolas
$\{w=u+iv\colon \frac{u^2}{A^2}-\frac{v^2}{B^2}=1\}$ with $A^2+B^2=1$ and find $A=A(\theta)$ and $B=B(\theta)$.

(c) Find to what domain this  function  maps the unit disk  $\mathbb{D}=\{z\colon |z|<1\}$.

(d) Draw both domains.

(e) Check if the correspondence is one-to-one.

12
Final Exam / FE-P1
« on: December 18, 2018, 06:11:23 AM »
(a) Decompose into Taylor series at $0$ function $$f(z)=\frac{1}{z^2+2z+2}.$$ Find the radius of convergence $r$. Determine if the series is converging at $|z|=r$ (consider all points $z$ satisfying $|z|=r$).

(b) Decompose into Laurent's series at $\infty$ the same function. Also find the radius $R$ (so it converges as $|z|> R$).
 Determine if the series is converging at $|z|=R$ (consider all points $R$ satisfying $|z|=R$).


Hint: Represent $f(z)$ as the sum of functions of the form $\frac{a}{b+z}$.


13
Final Exam / FE-P6
« on: December 14, 2018, 08:06:54 AM »
Typed solutions only. Upload only one picture (a general phase portrait; for general one can use computer generated)
For the system of ODEs
\begin{equation*}
\left\{\begin{aligned}
&x'  = 2y(x^2+y^2+4)\, , \\
&y'  = -2x (x^2+y^2-16)
\end{aligned}\right.
\end{equation*}

(a) Find stationary points.

(b) Linearize the system at stationary points and sketch the phase portrait of this linear system.

(c) Find the equation of the form $H(x,y) = C$, satisfied by the trajectories of the nonlinear system.

(d)  Sketch the full phase portrait.

Hint: avoid redundancy: asymptotically (un)stable node, unstable node, stable center

14
Final Exam / FE-P5
« on: December 14, 2018, 08:03:41 AM »
Typed solutions only. Upload only pictures (at all stationary points on one picture and a general phase portrait  on another; for general one can use computer generated)

For the system of ODEs
\begin{equation*}
\left\{\begin{aligned}
&x'  = x(3x +2y -30)\, , \\
&y'  = y(2y-x-6)\,.
\end{aligned}\right.
\end{equation*}

(a) Describe the locations of all critical points.

(b) Classify their types (including whatever relevant: stability, orientation, etc.).

(c)  Sketch the phase portraits near the critical points.

(d)   Sketch the full phase portrait of this system of ODEs.

Hint: avoid redundancy: asymptotically (un)stable node, unstable node, stable center


15
Final Exam / FE-P4
« on: December 14, 2018, 07:55:46 AM »
Typed solutions only. No uploads

Find the general solution $(x(t),y(t))$ of the system of ODEs
\begin{equation*}
\left\{\begin{aligned}
&x' = x-2y + \sec(t)\, &&-\frac{\pi}{2}<t<\frac{\pi}{2},\\
&y' = x -\ \,y  \,.
\end{aligned}\right.
\end{equation*}
Hint: $\sec(t)=\frac{1}{\cos(t)}$.

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