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### Topics - Victor Ivrii

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1
##### Term Test 2 / Do not Post until Wednesday, 3--4 pm
« on: March 20, 2018, 08:18:45 PM »
At Wed, 3--4 pm I will post all problems, for all sittings.

Until the problem is posted, you solution is considered unauthorized and will be deleted. There is a student, who writes it today

2
##### Web Bonus Problems / Phaseportrait
« on: March 18, 2018, 12:43:58 PM »
Sketch the phaseportrait for the system below. Anyone can post a different solution, but there is a catch: it should be drawn with different s/w than already used and this s/w must be reported
\begin{align}
&x'= \sin(x)\cos(y)\\
&y'=-\cos(x)\sin(y) && -4\le x\le 4, \ -4\le y \le 4
\end{align}

3
##### MAT244--Misc / What s/w have you used for plots?
« on: March 18, 2018, 11:46:11 AM »
I see that some used s/w to plot which I am not familiar. Please post here!

4
##### Quiz-6 / Q6--T5101
« on: March 16, 2018, 08:19:34 PM »
a. Express the general solution of the given system of equations in terms of real-value functions.
b. Also draw a direction field, sketch a few of the trajectories, and describe the behavior of the solutions as $t\to \infty$.
$$\mathbf{x}' =\begin{pmatrix} -1 &-4\\ 1 &-1 \end{pmatrix}\mathbf{x}$$

5
##### Quiz-6 / Q6--T0901
« on: March 16, 2018, 08:17:56 PM »
a. Express the general solution of the given system of equations in terms of real-valued functions.
b. Also draw a direction field, sketch a few of the trajectories, and describe the behavior of the solutions as $t\to \infty$.
$$\mathbf{x}' =\begin{pmatrix} 4 &-3\\ 8 &-6 \end{pmatrix}\mathbf{x}$$

6
##### Quiz-6 / Q6--T0801
« on: March 16, 2018, 08:17:05 PM »
a. Express the general solution of the given system of equations in terms of real-valued functions.
b. Also draw a direction field, sketch a few of the trajectories, and describe the behavior of the solutions as $t\to \infty$.
$$\mathbf{x}' =\begin{pmatrix} 1 &-1\\ 5 &-3 \end{pmatrix}\mathbf{x}$$

7
##### Quiz-6 / Q6--T0701
« on: March 16, 2018, 08:16:03 PM »
a. Express the general solution of the given system of equations in terms of real-valued functions.
b. Also draw a direction field, sketch a few of the trajectories, and describe the behavior of
the solutions as $t\to \infty$.
$$\mathbf{x}' =\begin{pmatrix} 1 &2\\ -5 &-1 \end{pmatrix}\mathbf{x}$$

8
##### Quiz-6 / Q6--T0601
« on: March 16, 2018, 08:14:20 PM »
a. Express the general solution of the given system of equations in terms of real-valued functions.
b. Also draw a direction field, sketch a few of the trajectories, and describe the behavior of
the solutions as $t\to \infty$.
$$\mathbf{x}' =\begin{pmatrix} 4 &-3\\ 8 &-6 \end{pmatrix}\mathbf{x}$$

9
##### Quiz-6 / Q6--T0401
« on: March 16, 2018, 08:11:48 PM »
a. Express the general solution of the given system of equations in terms of real-valued functions.
b. Also draw a direction field, sketch a few of the trajectories, and describe the behavior of the solutions as $t\to \infty$.
$$\mathbf{x}' =\begin{pmatrix} 3 &-2\\ 2 &-2 \end{pmatrix}\mathbf{x}$$

10
##### Quiz-6 / Q6--T0301
« on: March 16, 2018, 08:10:13 PM »
a. Express the general solution of the given system of equations in terms of real-valued functions.
b. Also draw a direction field, sketch a few of the trajectories, and describe the behavior of the solutions as $t\to \infty$.
$$\mathbf{x}' =\begin{pmatrix} 3 &-2\\ 4 &-1 \end{pmatrix}\mathbf{x}$$

11
##### Quiz-6 / Q6--T0201
« on: March 16, 2018, 08:09:14 PM »
a. Express the general solution of the given system of equations in terms of real-valued functions.
b. Also draw a direction field, sketch a few of the trajectories, and describe the behavior of the solutions as $t\to \infty$.
$$\mathbf{x}' =\begin{pmatrix} -2 &1\\ 1 &-2 \end{pmatrix}\mathbf{x}$$

12
##### Quiz-6 / Q6--T0101
« on: March 16, 2018, 08:08:07 PM »
a. Express the general solution of the given system of equations in terms of real-valued functions.
b. Also draw a direction field, sketch a few of the trajectories, and describe the behavior of the solutions as $t\to \infty$.

$$\mathbf{x}' =\begin{pmatrix} 2 &-5\\ 1 &-2 \end{pmatrix}\mathbf{x}$$

13
##### Quiz-6 / Quiz 6 T5102
« on: March 16, 2018, 06:35:33 AM »
Solve
\begin{align*}
& \Delta u:=u_{xx}+u_{yy}=0&& \text{in }  r> a\\[3pt]
& u|_{r=a}=f(\theta),\\[3pt]
& \max |u| <\infty.
\end{align*}
where we use polar coordinates $(r,\theta)$ and f(\theta)=\left\{\begin{aligned} &1 &&0<\theta<\pi\\ -&1 &&\pi<\theta<2\pi. \end{aligned}\right.

The expected answer: solution as a series.

14
##### Technical Questions / Scans and snapshots
« on: March 11, 2018, 07:45:46 AM »
Some student posted a snapshot of the solution, and lamented that I rejected it as unreadable. I think I need to clarify:

Indeed, it is a high resolution snapshot, and while downloaded looks passable. But I find it unacceptable that to see it  properly, people need to download it. And in forum it does not fit into window (when you click on it) exactly because of its high resolution (lots of pixels). And both snapshots take 360KB, which is way too much. Imagine,  if the scan of a book of 200 pages was made in a similar way, it would be 36 MB. And most of these 36 MB would be wasted on the yellow colour of the pages. To make things worse your pages are not strictly horizontal.

You see, your smartphone is not a scanner, which in the proper s/w setup would make it black-white (bitonal), not even grayscale, leave alone colour, and would make pages properly oriented. Special s/w for smartphone can mitigate these shortcomings.

(This is why I ask to send me scans of doctor's notes rather than snapshots).

15
##### Quiz-5 / Q5--T0501, T5101
« on: March 09, 2018, 05:57:36 PM »
a. Transform the given system into a single equation of second order.

b. Find $x_1$ and $x_2$ that also satisfy the given initial conditions.

c. Sketch the graph of the solution in the $(x_1,x_2)$-plane for $t \ge 0$.

\left\{\begin{aligned} & x'_1= x_1 - 2x_2, &&x_1(0) = -1,\\ &x'_2= 3x_1 - 4x_2, &&x_2(0) = 2. \end{aligned}\right.

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