Show Posts

This section allows you to view all posts made by this member. Note that you can only see posts made in areas you currently have access to.


Messages - Victor Ivrii

Pages: 1 ... 4 5 [6] 7 8 ... 152
76
Chapter 9 / Re: the stability characteristics of all periodic solutions
« on: December 16, 2019, 03:38:04 PM »
Limit cycles (not circles) are not covered by final exam. In contrast to spiral point these cycles have two sides: external and internal. See picture

77
Term Test 2 / Re: Problem 4 (noon)
« on: November 24, 2019, 11:04:24 AM »
What everybody is missing

we see that characteristic roots $k_{1,2}=-1\pm \sqrt{2}i$ are complex, with negative real part. So, it is  stable focus  and with  clock-wise  orientation  since the bottom-left element is negative.

78
Term Test 2 / Re: Problem 4 (morning)
« on: November 24, 2019, 11:00:36 AM »
What everybody is missing:

we see that characteristic roots $k_{1,2}= \pm \sqrt{8}i$ are purely imaginary. So, it is  center  and with  counter-clock-wise  orientation  since the bottom-left element is positive.

79
Term Test 2 / Re: Problem 4 (main sitting)
« on: November 24, 2019, 10:45:37 AM »
What everybody is missing

it is  unstable focus  and with  clock-wise  orientation  since the bottom-left element is negative.


80
Term Test 2 / Re: Problem 3 (noon)
« on: November 24, 2019, 10:00:42 AM »
What everybody is missing
In problem got lost "classify point $(0,0)$"

saddle

81
Term Test 2 / Re: Problem 3 (morning)
« on: November 24, 2019, 09:55:08 AM »
What everybody is missing
In problem got lost "classify point $(0,0)$"

stable improper node; since the bottom left element is negative, it is clockwise

82
Term Test 2 / Re: Problem 3 (main sitting)
« on: November 24, 2019, 09:41:32 AM »
What everybody is missing:
Part of the problem "classify fixed point $(0,0)$".
It is unstable node,

83
Term Test 2 / Re: Problem 1 (noon)
« on: November 24, 2019, 08:41:08 AM »
$$
\boxed{  y= \Bigl(-\frac{1}{2}\ln (e^{2t}+1)+c_1 \Bigr)e^{t} + \Bigl( \arctan (e^t)+c_2\Bigr)e^{2t}. }
$$
 and
$$
\boxed{  y= \Bigl(-\frac{1}{2}\ln (e^{2t}+1)+\frac{1}{2}\ln (2)  \Bigr)e^{t} + \Bigl( \arctan (e^t)-\frac{\pi}{4}\Bigr)e^{2t}. }
$$

84
Term Test 2 / You may post solutions
« on: November 19, 2019, 04:25:47 AM »
After an (almost) perfect solution is posted, no need to post the same solution

85
Term Test 2 / Problem 4 (noon)
« on: November 19, 2019, 04:24:35 AM »
Find the general real solution to
$$
\mathbf{x}'=\begin{pmatrix}
1 & 3\\
-2 &-3\end{pmatrix}\mathbf{x}
$$
and sketch trajectories.

86
Term Test 2 / Problem 4 (morning)
« on: November 19, 2019, 04:23:46 AM »
Find the general real solution to
$$
\mathbf{x}'=\begin{pmatrix}
2 & -3\\
4 &-2\end{pmatrix}\mathbf{x}
$$
and sketch trajectories.

87
Term Test 2 / Problem 4 (main sitting)
« on: November 19, 2019, 04:23:20 AM »
Find the general real solution to
$$
\mathbf{x}'=\begin{pmatrix}
3 & 3\\
-2 &-1\end{pmatrix}\mathbf{x}
$$
and sketch trajectories.

88
Term Test 2 / Problem 3 (noon)
« on: November 19, 2019, 04:22:43 AM »
(a) Find the general solution of
$$
\mathbf{x}'=\begin{pmatrix} 1 &2\\
1 &0\end{pmatrix}\mathbf{x}
$$
and sketch trajectories.

(b) Find the general solution
$$
\mathbf{x}'=\begin{pmatrix} 1 &2\\
1 &0\end{pmatrix}\mathbf{x}+
\begin{pmatrix} 0 \\[1pt]
\dfrac{6 e^{3t }}{e^{2t}+1}\end{pmatrix}.$$

89
Term Test 2 / Problem 3 (morning)
« on: November 19, 2019, 04:21:57 AM »
(a) Find the general solution of
$$
\mathbf{x}'=\begin{pmatrix} -2 &1\\
-1 &0\end{pmatrix}\mathbf{x}
$$
and sketch trajectories.

(b) Find the general solution
$$
\mathbf{x}'=\begin{pmatrix} -2 &1\\
-1 &0\end{pmatrix}\mathbf{x}+
\begin{pmatrix} 0  \\
\dfrac{e^{-t}} {t^2+1} \end{pmatrix}.
$$

90
Term Test 2 / Problem 3 (main sitting)
« on: November 19, 2019, 04:21:01 AM »
(a) Find the general solution of
$$
\mathbf{x}'=\begin{pmatrix} 1 &1\\
-2 &4\end{pmatrix}\mathbf{x}
$$
classify fixed point $(0,0)$ and sketch trajectories.

(b) Find the general solution
$$
\mathbf{x}'=\begin{pmatrix} 1 &1\\
-2 &4\end{pmatrix}\mathbf{x}+
\begin{pmatrix} \dfrac{e^{4t }}{e^{2t}+1} \\
0\end{pmatrix}.
$$

Pages: 1 ... 4 5 [6] 7 8 ... 152