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### Messages - Victor Ivrii

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1321
##### Term Test 1 / Re: TT1--Problem 4
« on: February 14, 2013, 05:40:25 AM »
$\renewcommand{\Re}{\operatorname{Re}}$

Changyu -- wrong signs in the last 2 terms. Then your solution would collapse to $\cos(2x)+x\sin(2x)$ rather than to $\cos(2x)-x\sin(2x)$ as your solution. Definitely if you going through complex form you need to work only with half of roots so you look for
\begin{equation*}
\Re \bigl(C_1 e^{2ix} + C_2 x e^{2ix}\bigr).
\end{equation*}

Because of initial data, the real form is preferable but there is more. Note $y'(0)=y'''(0)=0$ and equation contains only even derivatives. Therefore if $y(x)$ is a solution, $y(-x)$ is also a solution and since solution is unique we conclude that $y(x)=y(-x)$ so it is an even function i.e.
\begin{equation*}
y(x) = C_1 \cos(2x) +C_3 x\sin(2x)
\end{equation*}
and now everything goes very fast: $y(0)=C_1$, $y''(0)=-4C_1+4 C_3$.

Devin, each approach has its own advantages and disadvantages and you need to be comfortable with both. Also note that $\cos(x)$ and $\sin(x)$ are not only real but also even and odd respectively which often makes life easier; $\cosh(x)$ and $\sinh(x)$ are used by the same reason and sometimes they give you an edge over pair $e^x$ and $e^{-x}$.

1322
##### Term Test 1 / Re: TT1--Problem 3
« on: February 14, 2013, 05:15:11 AM »
Devin, for me WolframAlpha gave exactly your answer for homogeneous equation. Probably you asked it not politely enough

PS Usage of double dollars in LaTeX is deprecated. There is command \Re but out of the box it returns $\mathfrak{R}$ so I redefined it
Code: [Select]
$\renewcommand{\Re}{\operatorname{Re}}$
(dollars needed to tell MathJax to pay attention, in normal LaTeX they would be wrong)

Marcia, I decided that variation of parameters deserves a reward. Note however that you made small computational  errors

1323
##### Term Test 1 / Re: TT1--Problem 2
« on: February 14, 2013, 05:00:19 AM »
See my comments to Problem 1. Waiting for typed solution.

1324
##### Term Test 1 / Re: TT1--Problem 1
« on: February 14, 2013, 04:55:35 AM »
I decided to be generous and awarded karma to all 4. Marcia definitely realized that her first scan (actually snapshot) was almost completely useless and reposted with double resolution; honestly, even her 2nd snapshot is inferior. Matthew positioned paper in the best possible way, and Yook used grayscale (better than colour; however black and white would be even better but it requires more knowledge--see my avatar for b/w) and a monster-resolution picture (but because it was grayscale file size was not much larger!)

Alexander' post is far superior (orientation is not that important, major advantage it is typed and could be easily edited and recycled so in the most strict approach (the first gets all) Matthew and Alexander would get their karma.

1325
##### 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*}

1326
##### 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*}

1327
##### 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 $y_2(\pi/2)=0$.

1328
##### 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*}

1329
##### Term Test 1 / Posting solutions
« on: February 13, 2013, 04:55:52 PM »
About 22:30 today  I will post problems--and then you post solutions. Please note that not only solutions but also discussion of solutions are welcome.

1330
##### MAT 244 Misc / Re: Will there be a lecture on Wednesday?
« on: February 12, 2013, 02:46:08 AM »
Will it be new material or reviewing old material?

A mixed bag

1331
##### Ch 3 / Re: Variation of Parameters
« on: February 12, 2013, 02:45:27 AM »
In equation (28) of theorem 3.6.1, it says that "where t0 is any conveniently chosen point in I".

This is for the integral limits for u1 and u2.
I don't understand how we can just conveniently choose some t0 for the general solution when it's not an initial value problem.

Looking for a general solution we can select initial point as we please; it definitely affects other constants. Look at
\begin{equation*}
\int_{t_0}^f f(t)\,dt'+C_0=\int_{t_1}^f f(t)\,dt'+C_1
\end{equation*}
as long as $C_1-C_0=-\int_{t_0}^{t_1}f(t')\,dt'$ and you can select any initial point $t_0$.

1332
##### Technical Questions / Re: Preview
« on: February 11, 2013, 07:09:56 PM »
However I don't think you are correct about Chrome: Preview works for me in Chrome and it is Webkit. I just checked with Opera--Preview works, and with Maxthon--Preview works too!

I didn't say that it doesn't work---I said that it reloads the page when you click "Preview". If everything were working properly then it wouldn't have to reload the page, but would insert the preview directly into the current page. This is evidence that none of the browsers really do what they're supposed to do. Only Firefox has a weird bug that causes the preview not to work at all. Even still, fixing the XML would improve the user experience on all browsers.

I checked: out of the box SMF does not seem to reload page but inserts preview, and with MJ hookup it reloads. I am afraid it is unavoidable.

Unfortunately there is no real interaction between MJ and SMF developers. Davide Cervone, lead developer of MJ, who is usually extremely helpful, wrote that he knows very little about SMF.

On the other hand, mediawiki even without MJ seems to always reload page for preview and AFAIK nobody objects (I agree: reloading is an annoyance but rather minor).

1333
##### MAT244 Announcements / Re: Tests (all sections)
« on: February 11, 2013, 06:56:33 PM »
For TT1, MT and TT2:

The night lectures start as usual and continued without any break until 30-25 min before test; then we go to EX