$(2x^2+1)y''-4xy'+4y=0$  $y_1(x) = x$ is given and I have to find $y_2(x)$ and solve equation. thanks.
I need help solving this past exam question, the professor posted exam questions with out solutions to last year's exam, I tried reduction of order, however that seemed to complicate things. And then I tried Euler-Cauchy method however I realized that won't work due to the $(2x^2 + 1)y''$.  Given $y_1=x$ is a solution $$(2x^2 + 1)y'' − 4xy' + 4y = 0$$ Find the second solution $y_2$ for equation.
Suppose we have a set of primes $p_1,\dots,p_t$. Prove that  $\log p_1,\dots,\log p_t$ is linear independent over $\mathbb{Q}$. Now, this implies $ \sum_{j=1}^{t}x_j\log(p_j)=0 \iff x_1=\dots=x_t=0$.  I think I have to use that fact that every $q\in\mathbb{Q}$ can be written as $\prod_{\mathcal{P}}$, where $n_p$ is a unique sequence ($n_2$,$n_3$,$\dots$) with domain $\mathbb{Z}$. Here, $\mathcal{P}$ denotes the set of all integers.  Now how can I use this to prove the linear independency? Cross product term in linear regression why is it zero I'm trying to see why this is the case. I have taken  the LHS and added $X\hat{\beta}$ and subtracted $X\hat{\beta}$.  From this I can get both terms on the right but I end up with a cross product term  $2(y-X\hat{\beta})^T(X\hat{\beta} - X\beta)$  that I can't prove equals to zero. I think there is some trick I'm missing, anybody remember it? My local hackerspace has a CNC machine.  I can give them my Gerbers and they can print out the PCB.  The machine does not plate VIAs.  I need to do this after the CNC machine is finished.  What is the best process and materials to use? Is is true &lt;h1&gt; tag should not have more than 1 on every page for SEO?   Every pages on the header, it includes and &lt;h1&gt; tag. Website Name which show an image from logo class, like this:  &lt;div id="header"&gt;   &lt;h1 class="logo"&gt;      &lt;a href="#"&gt;Website Name&lt;/a&gt;   &lt;/h1&gt; &lt;/div&gt;   and I also include &lt;h1&gt; on the contact page, help, etc and article page for example:  &lt;h1&gt;Contact Us&lt;/h1&gt; &lt;h1&gt;Name of the Article Title here&lt;/h1&gt;   On the homepage, it just has one &lt;h1&gt; for the logo Site Name... other pages have 2 &lt;h1&gt; tags Encountered an error while updating through Synaptic Manager. Subsequent updates gave the error at the bottom of the output. I'm a noob so any help is appreciated.  al@al:~$ sudo apt-get -f upgrade Reading package lists... Done Building dependency tree        Reading state information... Done Correcting dependencies... Done The following NEW packages will be installed:   libjack-jackd2-0:i386 The following packages have been kept back:   linux-headers-generic skype 0 upgraded, 1 newly installed, 0 to remove and 2 not upgraded. Need to get 0 B/202 kB of archives. After this operation, 493 kB of additional disk space will be used. Do you want to continue [Y/n]? y (Reading database ... 310056 files and directories currently installed.) Unpacking libjack-jackd2-0:i386 (from .../libjack-jackd2-0_1.9.8~dfsg.2-1precise1_i386.deb) ... dpkg: error processing /var/cache/apt/archives/libjack-jackd2-0_1.9.8~dfsg.2-1precise1_i386.deb (--unpack):  './usr/share/doc/libjack-jackd2-0/buildinfo.gz' is different from the same file on the system dpkg-deb: error: subprocess paste was killed by signal (Broken pipe) Errors were encountered while processing:  /var/cache/apt/archives/libjack-jackd2-0_1.9.8~dfsg.2-1precise1_i386.deb E: Sub-process /usr/bin/dpkg returned an error code (1) Are the proofs for the properties of parallel lines, and that a triangle has 180 degrees, inherently tautological? I've noticed something while trying to prove the properties of parallel lines, and the properties that a triangle has 180 degrees. To prove the properties of parallel lines, such as alternate angles, you need to use the property that a triangle has 180 degrees. To prove a triangle has 180 degrees however, you need to use the properties of parallel lines. This really bothers me because of how circular it is. They are both reliant on each other to be true, and do not logically show, without being reliant on each other, why triangles have 180 degrees, and why parallel line properties are true.   So what I hoping for here is a way to prove parallel line properties without using the fact that a triangle has 180 degrees, or a way to prove triangles have 180 degrees without using parallel line properties. This way, things will be logical to me, and make sense.   Thanks in advance. Algebraic closure of $\mathbb{C}(x)$ is isomorphic to $\mathbb{C}$ Let $x$ be transcendental over $\mathbb{C}$. Let $K$ be the algebraic closure of $\mathbb{C}(x)$. How to show that $K$ is isomorphic to $\mathbb{C}$? Self-deletion and un-deletion without any changes should reinstate the original close/delete votes Rows to column conversion of file Suppose I have a file:  File1:   PAPER  TEAM  MANISH NISHA GARIMA JYOUTI ........etc    File2 I want:  PAPER     TEAM MANISH NISHA GARIMA     JYOUTI   Rows to column conversion of File1. Valid characters in a Java class name What characters are valid in a Java class name? What other rules govern Java class names (for instance, Java class names cannot begin with a number)? How can you tell when a sunset is going to be spectacular? I have a spot that I like to take sunset pictures. I find that the most spectacular moments last for only maybe 5-10 minutes. What I would like to do is know about a half hour before that a sunset is going to be spectacular, so I can grab my camera, tripod, walk 5-10 minutes to my sunset spot, set up, and have a few minutes to spare in case something happens. However, I've often missed really cool sunsets, because I just don't know when to photograph them. How can you tell beforehand when a sunset is going to be spectacular? Thanks! I have the two PGF nodes foo and bar positioned in a row.  \documentclass{article} \usepackage{tikz} \usetikzlibrary{positioning} \begin{document}   \begin{tikzpicture}[node distance=0.2cm,mynode/.style={rectangle,draw}]     \node[mynode] (foo)                {foo};     \node[mynode] (bar) [right=of foo] {bar};   \end{tikzpicture} \end{document}   How can I create a third node below those two nodes with the width of foo, bar and the node distance between them? Is there a way to do this with the fit library or is this the wrong approach? How do I get attention for old, unfixed bug reports and feature requests without official responses here on Meta? Delay jquery hover event? I would like to delay a hover event in jquery.  I am reading from a file when user hovers over a link or label.  I don't want this event to occur immediately in case the user is just moving the mouse across the screen.  Is there a way to delay the event from firing?  Thank you.  Example code:  $(function() {     $('#container a').hover(function() {         $('&lt;div id="fileinfo" /&gt;').load('ReadTextFileX.aspx',             {filename:'file.txt'},             function() {                 $(this).appendTo('#info');             }          );     },         function() { $('#info').remove(); }     }); });   UPDATE: (1/14/09) After adding the HoverIntent plugin the above code was changed to the following to implement it.  Very simple to implement.  $(function() {     hiConfig = {         sensitivity: 3, // number = sensitivity threshold (must be 1 or higher)         interval: 200, // number = milliseconds for onMouseOver polling interval         timeout: 200, // number = milliseconds delay before onMouseOut         over: function() {             $('&lt;div id="fileinfo" /&gt;').load('ReadTextFileX.aspx', {filename:'file.txt'},                 function() {                    $(this).appendTo('#info');                 }              );         }, // function = onMouseOver callback (REQUIRED)         out: function() { $('#info').remove();  } // function = onMouseOut callback (REQUIRED)     }     $('#container a').hoverIntent(hiConfig) } Hi i want to put this simbol "%" in the tick labels only in the x axis how can i do? We have been using this result without proof in my class, but I don't know how to prove it. Could someone point me in the right direction?  $$|1-e^{i\theta}|\le|\theta|$$  I believe this is true for all $\theta\in\mathbb{R}$. It is easy to show that the left side is bounded by 2 (triangle inequality), but I'm stuck otherwise. How to undelete a directory that was just deleted with "rm -R" command (in cloud drive)? I deleted an important directory with rm -R command. This is an very populer host data. It was located /var/zpanel/hostdata/ayrintilih After remove ayrintilih directory has gone everything.  I tried to undelete with ext4magic program. But it need to another hdd volume for recovery.  Error message:   ERROR: can not use "/root/recoveried" for recover directory. It's the same filesystem : "/dev/dm-0"   My ISP just replied my request:  Unfortunately there is no way to add another Volume to a virtual Instance.  If it would be a dedicated Server, we could do it, but not in a Cloudserver.  You could try to minimize the partition and create another one, that would  count as another Volume. We also recommend to have allways a  Backup for your Important files. Prime factorization of square numbers Let n be a natural number with unique prime factorization $p^m$... $q^k$ . Show that n can be written as a square if and only if  all (m, ...k) are even I came to this conclusion: the primes that can't be written as the sum of two squares (lets use$p\prime$ to denote) can be written as ${p}\prime \equiv 3 \;(\bmod\; 4)$.  I need a simple proof to show this and am stuck. I would appreciate guidance! Proof that a sequence of continuous functions $(f_n)$ cannot converge pointwise to $1_\mathbb{Q}$ on $[0,1]$