Topics

31

Term Test 1 / Problem 1 (noon)

« on: October 23, 2019, 05:54:36 AM »

(a) Find integrating factor and then a general solution of ODE
\begin{equation*}
\bigl(2y+y^2\sin(x)\bigr) + \bigl(\sin(2x)+2y\cos(x)\bigr) y'=0
\end{equation*}

(b) Also, find a solution satisfying $y(\dfrac{\pi}{4})=\sqrt{2}$.

32

Term Test 1 / Problem 1 (morning)

« on: October 23, 2019, 05:53:12 AM »

(a) Find integrating factor and then a general solution of ODE
\begin{equation*}
\bigl(-y\sin(x)+y^3\cos(x)\bigr) + \bigl(3\cos(x)+5y^2\sin(x)\bigr) y'=0
\end{equation*}

(b) Also, find a solution satisfying $y(\dfrac{\pi}{4})=\sqrt{2}$.

33

Term Test 1 / Problem 1 (main sitting)

« on: October 23, 2019, 05:51:05 AM »

(a) Find integrating factor and then a general solution of ODE
\begin{equation*}
\bigl(y +3 y^2e^{2x}\bigr) + \bigl(1+2ye^{2x}\bigr) y'=0
\end{equation*}

(b) Also, find a solution satisfying $y(0)=1$.

34

Term Test 1 / Please post solutions

« on: October 20, 2019, 06:25:41 PM »

You may post solutions to Test 1. All solutions posted before this announcement were removed (all of them seem to be  prepared in advance).
One user who was too smart for his/her own good (posted several times something like a single digit to stake out the lot) got a posting ban.

Please type sin, cos, log, ...  as \sin, \cos , \log , ... to produce $\sin (x)$,... (upright and with a proper horizontal spacing)

35

Quiz-1 / MOVED: MAT24f4 TUT5103 Quiz2

« on: October 07, 2019, 05:32:24 AM »

This topic has been moved to Quiz-2.

http://forum.math.toronto.edu/index.php?topic=1865.0

36

Quiz-1 / MOVED: MAT24f4 TUT5103 Quiz2

« on: October 07, 2019, 05:32:12 AM »

This topic has been moved to Quiz-2.

http://forum.math.toronto.edu/index.php?topic=1877.0

37

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

38

MAT244--Lectures & Home Assignments / LEC 5101-corrections to Lecture of Sept. 10

« on: September 11, 2019, 04:00:52 AM »

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*}

39

Technical Questions / emoji

« on: December 20, 2018, 08:37:39 PM »

😈 👿 👹 👺 💀 👻 👽 🤖 💩 😺 😸 😹 😻 😼 😽 🙀 😿 😾

40

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.

41

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

42

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 $z$.

43

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.

44

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.

45

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 $\frac{a}{b+z}$.