The PDE is given is: $$ 4u_{xx} + 5u_{xy} + u_{yy} + u_x + u_y = 2$$ I need to reduce this hyperbolic equation into a canonical form. I got the characteristics as $\xi = y-x$ and $\eta = y - \frac{x}{4}$  Let $w(\xi,\eta) = u(x(\xi,\eta),y(\xi,\eta)) $. After substituting the partial derivatives of $u$ as pd's of $w$, I worked out the canonical form as (if anyone can confirm my answer it would be reassuring) $$ w_{\xi\eta} - \frac{w_{\eta}}{3} = -\frac{8}{9}$$ How to solve this PDE to obtain the general solution?  I proceeded in the following way: $v = w_{\eta}$, which gives $$v_{\eta} - \frac{v}{3} = -\frac{8}{9} $$ And solving this equation I got $$  w(\xi,\eta) = \int 3e^{(\xi + g(\eta))/3} \ \partial \eta + \frac{8}{3}\eta + C$$ or  $$u(x,y) = \boxed{ C + \frac{8}{3}\left(y-\frac{1}{4}x \right) + \int 3e^{(y-x + g(y-x/4))/3} \ (\partial y - \frac{1}{4}\partial x) } $$ I am not sure about the result. It looks messy and I don't know if my approach is correct. Any input is appreciated. Thanks in advance.
How do I solve the following second order partial differential equation?  $4u_{xx}+5u_{xy}+u_{yy}+u_x+u_y=2$  I have classified the equation to be hyperbolic and changed variables to obtain the canonical form as $u_{\epsilon\nu}={1\over{3u_\nu}}-{8\over9}$ which I believe is correct but I am struggling to find the general solution? A step by step solution would be appreciated.
Complete classification of the groups for which converse of Lagrange's Theorem holds I see that there a number of questions (, , , , ) on best practices, none of them answer the issue I am going to talk about.  We all need to prepare large documents (or slides) from time-to-time. And once you have reached a certain number of pages, you find that compiling the whole document takes a substantial time, which you need to do to check that the part you have written is error free, specially if it contains a number of images.   To circumvent the situation, I have used the following practices over years to keep compile time at minimum.   Divide the document in to many chapters: The chapters are put in separate files. Only the one I am working one is uncommented.  \documentclass{book}  \begin{document}  %\input {chapterone.tex} \input {chaptertwo.tex} %\input {chapterthree.tex} %\input {chapterfour.tex}  \end{document}    The above technique can also be used for \include.   Keep a preamble replica of the main file in a temporary file (build file): I use this technique when I am preparing a presentation with many slides, say a hundred. Prepare a main file, copy it to another build file. Delete everything between  \begin{document} and \end{document} in the build file. Prepare a slide (or a group of slides) in the build file. Cut and paste the contents between \begin{document} and \end{document} in the build file to relevant location in the main file only when satisfied with the output (image locations, overfull boxes etc.)     We can hear your inputs about the best practices you practice when a you are preparing a document or presentation from scratch.    Now, I have failed to find a good answer in the case when you have finished the whole document or presentation and now you need tweaking the small things, may be edit a text here, add an image there. What will be your best practice?  Perhaps, again keeping only one chapter uncommented will be usable here. But what about large presentations? May be adding a \end{document} just after the point-of-change will be beneficial, but only if you can put that before midpoint of the file.  Commenting a part of the file will also work, but that is error-prone, at least for me. JSF returns blank/unparsed page with plain/raw XHTML/XML/EL source instead of rendered HTML output I have these two CREATE TABLE statements:   CREATE TABLE GUEST (   id int(15) not null auto_increment PRIMARY KEY,   GuestName char(25) not null );  CREATE TABLE PAYMENT (   id int(15) not null auto_increment   Foreign Key(id) references GUEST(id),   BillNr int(15) not null );   What is the problem in the second statement? It did not create a new table. Splash Screen waiting until thread finishes I still have a problem with the splash screen. I don't want to use the property SC.TopMost=true.  Now my application scenario is as follows:  in progeram.cs:  [STAThread] static void Main() {     new SplashScreen(_tempAL);// where _tempAL is an arrayList     Application.Run(new Form1(_tempAL)); }   in SplashScreen class:  public SplashScreen(ArrayList _Data) {     DisplaySplash() }  private void DisplaySplash() {     this.Show();     this.TopMost = true;     this.CenterToScreen();     this.SetTopLevel(true);      _allServerNarrators = new string[10];     for (int i = 0; i &lt; _allServerNarrators.Length; i++)         _allServerNarrators[i] = null;      GetFromServer();      this.Hide();     _serverData = new ArrayList();     _thisData.Add(_allServerNarrators);     _thisData.Add(_serverNarrators);  } private void GetFromServer() {     _serverNarrators = new ArrayList();     string _file = "Suras.serverNar";      if (!Directory.Exists("c:\\ASGAQuraan"))         Directory.CreateDirectory("c:\\ASGAQuraan");      while (counter &lt; 4 &amp;&amp; _serverFiles == null)     {         if (Download("c:\\ASGAQuraan", _ftpServerIP, _file))         {             StreamReader _strReader = new StreamReader                          ("c:\\ASGAQuraan\\"+_file,System.Text.Encoding.Default);             string _line = _strReader.ReadLine();             string _word;              while (true)             {                 while (_line != null)                 {                     _word = _line.Substring(0, _line.IndexOf("*"));                     int _narId = Convert.ToInt32(_word);                     _line = _line.Substring(2);                     int k = 0;                     _serverNarratorNode = new ArrayList();                     while (true)                     {                         int ind = _line.IndexOf("*");                         if (ind &gt; 0 &amp;&amp; ind &lt; _line.Length)                         {                             string str = _line.Substring(0, (ind));                             if (k == 0)                             {                                 _allServerNarrators[_narId] = str;                                 _serverNarratorNode.Add(str);                             }                             else                             {                                 _serverNarratorNode.Add(str);                             }                             _line = _line.Substring(ind + 1);                             k++;                         }                         else                         {                             _line = null;                             break;                         }                     }                     _serverNarrators.Add(_serverNarratorNode);                     _serverFiles = "added";                 }                 _line = _strReader.ReadLine();                 if (_line == null)                 {                     break;                 }             }         }         else             counter++;     } }   What I want is something in the splash screen class which waits until the thread finishes.  For more details, please tell me what I need to tell you. What exactly is the Box-Jenkins method for ARIMA processes? The  page says that Box-Jenkins is a method of fitting an ARIMA model to a time series. Now, if I want to fit an ARIMA model to a time series, I will open up SAS, call proc ARIMA, supply the parameters $p,d,q$ and SAS will give me AR and MA coefficients. Now, I can try different combinations of $p,d,q$ and SAS will give me a set of coefficients in each case. I select the set with the lowest Akaike information criterion.  My question is: where in the above procedure did I use Box-Jenkins? Am I supposed to use Box-Jenkins to come up with initial estimates of $p,d,q$? Or did SAS use it internally somehow? I am using extensively trigonometric functions when an angle is given in degrees. Some of these functions like sine or cosine have rational values, for example, the well   known example is that $\cos(\theta) =0.6 $ and $\sin(\theta) =0.8 $.       However besides the case of multiplicity of $90^\circ$ it seems there are no rational numbers $\theta$  with simultaneously rational  values of  sine and cosine.      Is it possible somehow to prove that for rational values of an angle given in degrees there are  no values simultaneously rational of sine and cosine functions, beside obvious case of multiplicities of $90^\circ$? How can I get to Iceland on a cargo ship? Planning a low budget and environment friendly trip to Iceland from Helsinki, Finland next summer for me and my friend. Haven't been able to find relevant information on possibility for cargo travel (from where - eg. Norway, Denmark? - timetables, prices etc). Expected departure from Helsinki around June 18th, return at latest on July 7th 2015. I'm very thankful for any related advice (and first time on stackexchange, so apologies if I didn't formulate my question in a best possible way) I'm from Tunisia and I got a Schengen visa from the Austrian Embassy for  28 days, the “duration of stay”, and the number of entries is 1. But the validity of the visa is from 02/08/2015 to 13/09/2015.  Do I have to stay only 28 days, for example from 02/08/2015 to 29/08/2015 or can I stay there until 12/09/2015 ?  And if I have got to stay only 28 days, then can I choose the days of stay, that's mean can I stay in Austria for example from 09/08/2015 to 05/09/2015? Let $n$ is a natural number. Prove that $$\left\lfloor\frac{n}{1}\right\rfloor+\left\lfloor\frac{n}{2}\right\rfloor+....+\left\lfloor\frac{n}{n}\right\rfloor+\left\lfloor{\sqrt{n}}\right\rfloor$$   is even.   I used induction and the inequality $$x-1&lt;\lfloor{x}\rfloor\le{x}$$ to prove it.  Is there any other, nicer way to do it? How is it possible in Eclipse JDT to convert a multiline selection to String. Like the following  From:  xxxx yyyy zzz   To:  "xxxx " + "yyyy " + "zzz"   I tried the following template   "${line_selection}${cursor}"+   but that way I only get the whole block surrounded not each line separately. How can I achieve a multiline processing like commenting the selected block? Why is plain-hash-then-encrypt not a secure MAC? It seems that even in MAC-then-encrypt systems like SSL, something like HMAC is used rather than a plain hash. Why?   Suppose we use some stream cipher; then why can't we use $Encrypt(m | H(m))$ as the MAC-then-encrypted version of the message? Assuming no bad relations between $Encrypt$ and $H$, what are the possible weaknesses? It seems that in this case we are encrypting the hash with part of a secretly keyed stream cipher keystream, and the hash is secure; does this not fulfil an approximation of the ideal MAC?  This must be somehow bad, since no secure protocol seems to use it. Why? subideals of an ideal If $R$ is a ring, J is an ideal in $R$, and $I$ is an ideal of $J$ (with $J$ considered as a ring), does it follow that $I$ is an ideal of $R$?  That is, is $I$ necessarily closed under multiplication by elements of $R$?  Surely $I$ is closed under multiplication by elements of $J$ (since $I$ is an ideal of $J$).  The "obvious" approach to proof fails so quickly that it must be false, that "an ideal of an ideal is not necessarily an ideal of the big ring".  Please provide a counterexample.  (Or a proof?) What methods of ‘clearfix’ can I use? Sounds of supersonic objects Would you, as a stationary observer, hear a sound in reverse when the source of the sound travels with twice the speed of sound? Of course, he wouldn't hear anything at all before the airplane passed by and the sound reaches him, but what about after? Does it reverse due to the Doppler effect? What happens if I combine Ur-priest and prestige class to exceed 10 effective spellcasting levels? Suppose I have a human character who invested 5 levels on Wizard, 2 on Ur-priest(by appropriate feat chain), and 8 levels of Mystic Theurge, net level of 15. This allows you to cast spells as stated in 10th level Ur-priest Spells per day table. Now, assume that I leveled up, and I choose Mystic Theurge as the next level class. This makes me to look up Ur-priest table for Level 11, which does not exist. In this case, what happens to my divine spells per day? Does it continue to progress "in the same pattern", as Hellfire Blast progresses with Hellfire Warlock-Legacy Champion build? Does it stop and I only get Wizard spells per day? Or, isn't it allowed to do so?  I am totally aware that although the answer is "yes", it does not mean it means spells over 9th level will be granted to my character unless I take Improved Spell Capacity, which requires me to be epic. What I am curious is the progression of spells per day of existing spells.  P. S. Don't ask me how to satisfy the requirements. This is possible by combining Apprentice (Criminal), Iron Will and Great Fortitude as level-up feat, and Spell Focus (evil) as a Wizard bonus feat (RAW, it is a "Spell Focus" feat, anyway.) I have few friends.   I have a few friends.      I thought "few" means just one, two or even none. "A few" typically means more than two. However it seems to me some people say "few" when they really mean "a few", am I right? Show that any set in a metric space can be written as the intersection of open sets Show that any set contained in the metric space $(X, d)$ can be   written as the intersection of open sets.   Definitions: A set $A \subseteq X$ is open if $\forall x \in A$, $\exists \varepsilon&gt;0$ such that $B_{\varepsilon}(x) \subseteq A$. A set $C \subseteq X$ is closed if and only if $X \setminus C$ is open.   I also know that the intersection of a finite collection of open sets is open and that any open ball contained in $X$ is open. How can I prove this question? How to prove Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$? Last activity on question shows modified by user who did not ask or answer .       The  has no sign of the user sharth The  also does not have any sign of related activity   I would like to know why the user's name is associated with the question. At least why their display name shows up as having modified the question timeline activity.  Possible reason that I can think of is that, .  Is this the reason?  Is there a way (for normal users), to look for such details, other than looking at the user activity and the question's revision history.