Hi the question I have to solve is:  $u_x + u_y + u = e^{x+2y} \ $  where $ u(x,0) = 0$  First, I tried to solve the question of the form:  $$ u_x + u_y + u = 0$$  We know that $\dot x = 1 \ $ and $\dot y = 1 \ $. From that we can introduce s where $$x(s) = s +x_0$$ and $$y(s) = s$$. Then if $z = u(x(s), y(s))$ we have that $\dot z + z = 0$.   By setting $s = y$ we have then that $x_0 = y-x$.   Then we can get that $u(x,y) = e^{-y} \ g(y-x)$.  Now, to get $\dot z + z = e^{x+2y}$ I am completely stuck
It is asked to solve the PDE  $$u_x + u_y + u = e^{x+2y}$$  My attempt: We have that $$\frac{dx}{1}=\frac{dy}{1}=\frac{du}{e^{x+2y}-u} $$  $\Rightarrow \frac{dy}{dx}=1 \Rightarrow y=x+C_1 \iff C_1=y-x$  $\Rightarrow \frac{du}{dx}= e^{x+2y}-u \iff \dot{u}+u=e^{x+2y} \iff \\ \frac{d}{dx}(e^x u) = e^{2x+2y} \iff u = e^{-x} ( \frac{1}{2}e^{2x+2y} + C_2)$  As $u$ is constant along its characteristics:  $$u(x,y)=\frac{e^{-x}}{2}(e^{2x+2y} + 2C_2(y-x))$$  where $C_2$ is any function of one variable. But wolfram gave the following answer  $$u(x,y)=\frac{e^{-x}}{4}(e^{2x+2y} + 4C_2(y-x))$$  as you can check   What am I doing wrong?
How to get the last argument to a /bin/sh function What's a better way to implement print_last_arg?  #!/bin/sh  print_last_arg () {     eval "echo \${$#}"  # this hurts }  print_last_arg foo bar baz # baz   (If this were, say, #!/usr/bin/zsh instead of #!/bin/sh I'd know what to do.  My problem is finding a reasonable way to implement this for #!/bin/sh.)  EDIT: The above is only a silly example.  My goal is not to print the last argument, but rather to have a way to refer to the last argument within a shell function.    EDIT2: I apologize for such an unclearly worded question.  I hope to get it right this time.  If this were /bin/zsh instead of /bin/sh, I could write something like this  #!/bin/zsh  print_last_arg () {     local last_arg=$argv[$#]     echo $last_arg }   The expression $argv[$#] is an example of what I described in my first EDIT as a way to refer to the last argument within a shell function.  Therefore, I really should have written my original example like this:  print_last_arg () {     local last_arg=$(eval "echo \${$#}")   # but this hurts even more     echo $last_arg }   ...to make it clear that what I'm after is a less awful thing to put to the right of the assignment.  Note, however, that in all the examples, the last argument is accessed non-destructively.  IOW, the accessing of the last argument leaves the positional arguments as a whole unaffected. For what values of $\alpha$, does this integral converge? For what values of $\alpha&gt;0$ this integral converges $$\int_0^{\infty} \frac {\sin x}{x^{\alpha}}dx $$ What causes an electric shock - Current or Voltage? Though voltage and current are two interdependent physical quantity, I would like to know what gives more "shock" to a person - Voltage or Current? In simple words, will it cause more "electric - shock" when the voltage is high or Current is more? I'm about to fly from Istanbul to Seoul and I have a useful amount of Turkish Lira (TRY) left but I don't have any South Korean Won (KRW) to pay for the metro to my hostel when I get there.  Several times in the past I've ended up being stuck with currencies that are very difficult to exchange in the country I've arrived in.  Can I buy KRW at the airport in Istanbul?  (I've also asked , since Stack Exchange questions are meant more for the long tail than just for the original asker.) Vertical alignment in tabularx X column type I'd like to create a new column type which works exactly like a X column from tabularx but that also has the text vertically centered. I tried with the newcolumntype command but I can't figure out where to put the second column type? Example:  \newcolumntype{n}{&gt;{\centering\arraybackslash}X}   Where should I put the mto specify that the cells content should be vertically centered? Adjust sound notification per chat room At the top right of a chat room, just above the room title, there is a speaker icon. Clicking on this icon brings up the settings for "Sound notifications":    Currently, a change to this setting affects all chat rooms. Selecting "visible room" means that an audio notification is given for any activity in any room that is currently open in a browser tab (regardless of whether that is the active tab).  It would be useful to be able to adjust audio notification settings for individual rooms. For example, The Nineteenth Byte is quite a busy room, so I'd like to be notified only when mentioned. The Cornell Box is a much quieter room, and I'd like to be notified any time there is activity regardless of whether I am mentioned. Currently I can set all rooms to notify me of any activity, or I can set all rooms to notify me only when mentioned, but I can't choose a different setting for each room.  This would also be particularly useful when setting up a room for a very specific purpose, such as discussion of a particular question. Stripboard/veroboard/matrix board design software I'm prototyping a simple layout for a small project on matrix board, but figured there must be a better way than drawing it all out on paper. Are there any aimed at matrix board?  I've found the following software for stripboard, but nothing explicitly for matrix board:     Stripboard Magic (seems to be abandonware, but there's page about it )  Stripboard Editor Limited free version, designed to integrate well with     Any more I've missed? Any recommendations? I'm having trouble getting basic APA bibliographies to work. The same thing works with nejm style, but changing the style to apa gives errors:  ! Undefined control sequence. &lt;argument&gt; \mkbibdateapalongextra  {year}{month}{day}\iffieldundef {endyear}{...  Package biblatex Warning: Macro 'name:apa:last-first' undefined. (biblatex)                Using \newbibmacro.  Package biblatex Warning: Macro 'name:apa:first-last' undefined. (biblatex)                Using \newbibmacro.   I'm using texlive 2012, as you can see I'm using biber, yes, I reran biber, and also tried delting the .bbl file and rerunning tools. My source markup is:  \documentclass[12pt]{article}  \usepackage[hidelinks]{hyperref} \usepackage[backend=biber,style=apa]{biblatex}  \addbibresource{BibResource.bib}  \begin{document}  test\parencite{ormrod2004human}  \clearpage  \printbibliography  \end{document} Recently I ran across the sentence:     "Just why the law prescribed thirty-nine lashes instead of forty or forty-one and so on, must needs remain unanswered."   How did a plural verb like "needs" wind up as an adverb?  Is it alone in this phenomenon, or are there other examples that appear to be an odd misplacement of a second verb, but are actually adverbs as well? Mutual Independence Definition Clarification Why can't Transporter technology be used as a weapon? Seems like this is the most powerful weapon of all: transport your enemy into a bulkhead, planet, etc. Transport your enemy's ship inside a planet, etc.  Is there some technical reason that it can not be used this way? Can we use media9 to show a video on a PDF but the video is not embedded to the PDF? I have video tutorials that I want to show in my presentation with beamer class. I don't want the video files get embedded to the PDF because they will make the PDF size very very huge.  If you know how the videos can be played on the HTML with a reference rather than embedding the video file to the HTML file, I want something like this to be applied to my PDF presentation. I just want to reference the video URL from my PDF presentation.   Shortly speaking, can we use media9 to show a video on a PDF but the video is not embedded to the PDF?  Uploading those videos to a web server might be difficult for me because internet connection is very slow and sometimes not available in the presentation room. :-) Abelian quotient group I'm stuck on the following practice problem.   Any hints would be appreciated.  Suppose $N$ is a normal subgroup of  $G$ such that every subgroup of $N$ is normal in $G$ and $C_{G}(N) \subset N$. Prove that $G/N$ is abelian.  I'm not sure how to use the fact that $C_{G}(N) \subset N$.  Thanks Error message in my excel formula I'm adding two footnotes to a section heading:  \documentclass{article} \usepackage[english]{babel} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{lipsum}  \usepackage[stable]{footmisc}  \usepackage{titlesec}  \begin{document} \section{test \footnote{This is footnote no1.} \footnote{This is footnote no2.} } \lipsum[4] \end{document}   The heading will have both of the footnotes, but at the end of the page only the second is displayed. The problem appears to be \usepackage{titlesec}, because it works fine without it.    Stuff I specified in \usepackage{titlesec}:  \renewcommand\thesection{\arabic{section}}  \renewcommand\thesubsection{\arabic{subsection}}  \renewcommand\thesubsubsection{\arabic{subsubsection}}  \titleformat{\section}[block]{\Large\scshape\centering}{\thesection}{1em}{}  \titleformat{\subsection}[block]{\large\scshape}{\thesection.\thesubsection}{1em}{}  \titleformat{\subsubsection}[block]{\large\scshape}{\thesection.\thesubsection.\thesubsubsection}{1em}{}   Thanks in advance! Zsh get terminfo keys with modifiers Need help understanding "getbits()" method in Chapter 2 of K&R C In chapter 2, the section on bitwise operators (section 2.9), I'm having trouble understanding how one of the sample methods works.  Here's the method provided:  unsigned int getbits(unsigned int x, int p, int n) {     return (x &gt;&gt; (p + 1 - n)) &amp; ~(~0 &lt;&lt; n); }   The idea is that, for the given number x, it will return the n bits starting at position p, counting from the right (with the farthest right bit being position 0). Given the following main() method:  int main(void) {     int x = 0xF994, p = 4, n = 3;     int z = getbits(x, p, n);     printf("getbits(%u (%x), %d, %d) = %u (%X)\n", x, x, p, n, z, z);      return 0; }   The output is:     getbits(63892 (f994), 4, 3) = 5 (5)   I get portions of this, but am having trouble with the "big picture," mostly because of the bits (no pun intended) that I don't understand.  The part I'm specifically having issues with is the complements piece: ~(~0 &lt;&lt; n). I think I get the first part, dealing with x; it's this part (and then the mask) that I'm struggling with -- and how it all comes together to actually retrieve those bits. (Which I've verified it is doing, both with code and checking my results using calc.exe -- thank God it has a binary view!)  Any help? Decreasing font size in verbatim Two different solutions to integral How to prove that $p-1$ is squarefree infinitely often? How to prove that $p-1$ is squarefree infinitely often, where $p$ is prime?  I had thought of using Dirichlet's theorem on arithmetic progressions.  The number of squareful values of $p-1$ less than or equal to $n$ is bounded by $\sum_{q}{\pi_{q^2,1}(n)}$, each term $\pi_{q^2,1}(n) \approx \frac{\pi(n)}{q \cdot (q-1)}$, and $\sum_{q}{\frac{1}{q \cdot (q-1)}} \lt 1$.  But using for example , the possibility is open that the error in each approximation for $\pi_{q^2,1}(n)$ may continue to increase with $q$, and since the sum runs up to $\sqrt{n}$, this argument is invalid.  Is there a more refined version of Dirichlet's theorem that can be used to circumvent this issue?  Or an entirely different way to prove this statement?