$$x^2+y^2=\frac{dy}{dx}$$  Originally, this is a question I asked on Quora.     Maybe this question has been solved but I was wondering if it can be done by integrating factor?  $$x^2=\frac{dy}{dx}-y^2$$  Using method of integrating factor  $$\text{I.F}= e^{\int-ydx}$$  $$\text{I.F}= e^{-yx}$$  Can it be done in this way?
$$y'=x^2+y^2$$ I know, that this is a kind of Riccati equation, but is it possible to solve it with only simple methods? Thank you
R stem and leaf values How to make a stem and leaf plot with the following data in "R":  1717 1719 1645 3739 3024 3664 3830 2991 2430 2730 3469 5086 2119 3021 3292 2844 3426 2067 3215 2767 3124 2573 2840 2449 2584 1505 1390 1645 2497 3466 3228 3192  The only way to get a reasonable amount of stems is to make the stem the first digit in each number and the leaves will all be 3 digit numbers. I am not able to split the data in this way, R represents the stem with 1 digit but round my numbers  so 2584 is shown as 2| 5 (3 decimals to the right) which would be 2500. How do I adjust my stem and leaf numbers? proving how many permutations exist. Show that the number of permutations of n numbers, for n ≥ 2, with two cycles is at most (n−1)!logn. Unsure as to were to start any help would be appreciated. Using PTSerif-TLF for Cyrillic with TeX Gyre Pagella When should N-Gons be used, and when shouldn't they? Support for N-Gons (polygons consisting of more than 4 vertices) has been in Blender for a while now, but many of us still may not know when to use them, and how to do so effectively.  In what situations do N-Gons really come in handy? When should they be just plain left out? Do they have any place in finished models? Contour method to show that $\int_0^\infty\frac{x-\sin x}{x^3} \, dx=\frac\pi4$ Show that $$\int_{0}^{\infty} \frac{x - \sin(x)}{x^3} \, dx = \frac{\pi}{4}$$   My attempt is as follows: Let $$f(z) = \frac{z - i e^{iz}}{z^3}$$ and consider the contour on $[\epsilon, R]$ followed by a semicircular arc in the counter clockwise direction, then on $[-R, -\epsilon]$, then the semicircular clockwise contour avoiding the origin. We have, then, that  $$0 = \int_{\Gamma} f(z) dz = \int_{[\epsilon, R]} f(t) dt + \int_{C_R}f(Re^{it})Rie^{it}dt + \int_{[-R, -\epsilon]}f(t)dt + \int_{C_{\epsilon}}f(\epsilon e^{-it})\epsilon i e^{-it} dt$$  Then the first and third integrals ( $I_1$ and $I_3$) combine so that  $$I_1 + I_3 = 2\int_{\epsilon}^R \frac{t - \sin{t}}{t^3}\,dt$$  Further,   $$|I_{C_R}| \leq \int_0^\pi \left|\frac{Re^{it} - ie^{-R\sin{t}}e^{iRcos{t}}}{R^2 e^{2it}} \right|dt \rightarrow 0 \text{ as } R\rightarrow \infty$$  (I've omitted the details, it isn't too bad to bound)  However, I'm having trouble computing the limit  $$\lim_{\epsilon \rightarrow 0}\int_{C_{\epsilon}} f(\epsilon e^{-it})\epsilon i e^{-it} dt$$  No matter which way I look at it, it seems like this limit does not exist. Perhaps I'm seeing something wrong or have I chosen a bad $f(z)$? A question on $p$-groups, and order of its commutator subgroup. $\textbf{QUESTION-}$ Let $P$ be a p-group with $|P:Z(P)|\leq p^n$. Show that $|P'| \leq p^{n(n-1)/2}$.  If $P=Z(P)$ it is true. Now let $n &gt; 1$, then  If I see $P$ as a nilpotent group and construct its upper central series, it will end , so let it be,  $e=Z_0&lt;Z_1&lt;Z_2&lt;......&lt;Z_r=P$   Now as $Z_{i+1}/Z_i=Z(P/Z_i)$, so if if I take some $x\in Z_2$\ $Z_1$ then $N$={$[x,y]|y\in P$} $\leq Z_1(P)$ and $N \triangleleft P $, so $P/N$ is a group with order $\leq p^{n-1}$.  Now if I let $H=P/N$ then obviously |$H/Z(H)$|$\leq p^{n-1}$.  Now $H'\cong P'N/N \cong P'/(P' \cap N)$ so from here I could finally bring $P'$ atleast into the picture, now |$P'$|=$|H'|.|P'\cap N|$ so $|P'|\leq |H'||N|$.   This is where I am $\textbf{STUCK}$  Now , from here how can I calculate or find some power $p$ bounds on $|H'|$ and $|N|$ so i could get my result. I'd like to make links to all files in a directory tree. Which means, create the same directory structure and make the links in them to the respective subdirectory in the original dir.  That could be done with   cp -R -s ../foo .   But - that needs an absolute path. So rather  cp -R -s `readlink -e ../foo` `readlink -e .`   I would like the resulting links to be relative.  How would I do that? Would time freeze if you could travel at the speed of light? What is the difference between "I am done" and "I am through"? Some questions are getting closed as duplicates - good, the system is working. Unfortunately, some are getting closed as duplicates of other questions which are already deleted. This leaves all 1-9,999 reputation users left scratching their heads when they try to view the duplicate. Even worse, imagine being the author of the question and having your question closed as a duplicate when the original question isn't accessible.  I think something is wrong. Today I undeleted a question (left it closed) since it was referenced in the close-reason for another question. Comments indicated that people were clearly irritated that they could not access the duplicate     [Comment] Ok, who thought it was a good idea to close this as duplicate of a deleted question, which most people cannot even see? For the record, I'm reopening - clearly there's enough interest in this question. – Kobi 14 hours ago   ---   [Comment] The PosibleDuplicate link is broken. Voting to reopen. – Rahul G 7 hours ago   ---   [Flag] Closed as duplicate of a deleted question, fail. Thanks for moderating. – {username removed} 14 hours ago   My undelete-action was short-lived, as a few users returned to delete the post again.  It certainly cannot be desirable to reference posts that the vast majority of users cannot even see, especially when the author of the redundant question cannot see (at the time of writing this, the author has a reputation of 18).  I'm not sure what the solution is to this, but I'd like to hear some ideas.  Questions  Question:  Duplicate of: Ubuntu 11.10 is installed, and I have updated my system via Ubuntu Software Centre, and after the update 3 updates remain in the list. But I am unable to select them, the tickboxes cannot be checked and Ubuntu Software Centre gives no error when I try to check.  Also, below it says "There is no update to be installed" and even though there seems to be updates Install Updates button is unclickable.  What should I do in order to install these updates? I was wondering how to make a clickable terminal shortcut for my desktop using the command easytether connect.  I don't want to have to open/close the terminal every time I wish to connect to the Internet through my phone.  I've tried updating my gnome-panel but I can only make a new directory on my desktop. MikTeX 2.9, install packages on the fly? Removing Ubuntu from a dual-boot setup with Windows What is a "hat"? What exactly is a hat?  I have seen a lot of people arguing the significance of hats and the limited edition hats. I know the hats are sort of linked to the Winterbash stuff, but what exactly is a hat, what does it do, how do I get one?  Is there a full list of hats? SQL comma-separated row with Group By clause In the sentence-     You must wear a suit to an interview   shouldn't the to be replaced by for? Or what's the difference between these two here? How do I add Python to the Windows PATH? I want to be able to run Python commands from the Windows CMD. However, if I don't specify Python's full path for each command, I get an error saying "Python is not recognized as an internal or external command, operable program or batch file."  How do I add Python to the Windows PATH permanently? Is it possible to have two background images?  For instance, I'd like to have one image repeat across the top (repeat-x), and another repeat across the entire page (repeat), where the one across the entire page is behind the one which repeats across the top.  I've found that I can achieve the desired effect for two background images by setting the background of html and body:  html {     background: url(images/bg.png); }  body {     background: url(images/bgtop.png) repeat-x; }   Is this "good" CSS?  Is there a better method?  And what if I wanted three or more background images? How to create complex conditional clauses / queries? I've been reading the documentatoin on  and also studied the api description for  and  but I fail to create a $query with 3 -&gt;conditions:   select all between 0 and 100 but exclude 23, 48 and 97 select also 107, 123, 126 and 128   That's what I have tried so far:  $query-&gt;db_or()   -&gt;condition('my_field', array(107, 123, 126, 128), 'IN')   -&gt;condition(db_and()-&gt;condition('my_field', array(0, 100), 'BETWEEN')                       -&gt;condition('my_field', array(23, 48, 97), 'NOT IN'));   But I get the following error:  PHP Fatal error:  Call to undefined method SelectQuery::db_or() in [... first line]   I've tried to play around with the syntax like that:  $query-&gt;condition(-&gt;db_or()   -&gt;condition('my_field', array(107, 123, 126, 128), 'IN')   -&gt;condition(db_and()-&gt;condition('my_field', array(0, 100), 'BETWEEN')                       -&gt;condition('my_field', array(23, 48, 97), 'NOT IN')));   But again getting errors like that:  PHP Parse error:  syntax error, unexpected T_OBJECT_OPERATOR, expecting ')' in [... first line]   I also tried working with variables:  $cond01 = array('my_field', array(107, 123, 126, 128), 'IN'); $cond02 = array('my_field', array(0, 100), 'BETWEEN'); $cond03 = array('my_field', array(23, 48, 97), 'NOT IN'); $db_and = db_and(); $db_and-&gt;condition($cond02); $db_and-&gt;condition($cond03); $db_or = db_or(); $db_or-&gt;condition($cond01); $db_or-&gt;condition($db_and); $query-&gt;condition($db_or);   But this again yields drupal errors like that:  SQLSTATE[42S22]: Column not found: 1054 Unknown column 'Array' in 'where clause'   How to create a complex conditional database query with both AND and OR statements?