Determine if $x(t)$ is stable, where $x(t)$ is a solution to the differential equation:  $x'''+x''+4x'+4x = e^t(H(t) - H(t-3))$  and x(t) is defined as:  $x(t) = 0.1e^{-t} + 0.1e^t - 0.1\sin(2t) + 0.1e^3 (\sin(2(t-3)) + e^{-t+3}-e^{t-3})\times H(t-3)$  How do I approach this? Do I use eigenvalues? If so, I get 2i, - 2i and 1, is that correct and what do these values imply?  Any help is greatly appreciated!
Determine wether the differential equation is stable or not for the following conditions:  $x'''(0) = x''(0) = x'(0) = x(0) =0$  $\frac{d^3x(t)}{dt^3}+\frac{d^2x(t)}{dt^2}+4\frac{dx(t)}{dt}+4x(t) = e^t(H(t)-H(t-3))$  Now, I got that the solution to this differential equation reduces x(t) by use of Laplace Transforms, however, how do I determine whether or not it is stable?  $x(t) = 0.1e^{-t} + 0.1e^t - 0.1\sin(2t) + 0.1e^3 (\sin(2(t-3)) + e^{-t+3}-e^{t-3})*H(t-3)$  Now I first thought I would look at it graphically and I used Matlab software to get a plot as shown below:    And from the plot above it clearly appears to diverge (though this is not a proof).  Anyone have an idea as to an elegant proof to show the solution is unstable?  Anyone help would be greatly appreciated!
When GMing a game, how much should you prepare in advance? Obviously you need to have some high level idea of what's going on in the world around the players, and where you are heading with the story, but in an ongoing game at some point players will be players and will head off in a different direction to what you planned.  How far ahead should you be planning to minimise or eliminate being caught off guard, but also try to minimise the amount of 'wasted content' you come up with but never gets used? iPhone Apps section not showing up in iTunes Should the currently selected item in a combo/dropdown box also be included in the list of choices? For example, you have a combo box which contains 10 elements including the one that is currently selected. When you prompt to drop down the list to choose another, should the currently selected item also be in this list? StackExchange drop down hiding chat, blog and log out, concerns I am trying to understand an example from my textbook.  Let's say $Z = X + Y$, where $X$ and $Y$ are independent uniform random variables with range $[0,1]$. Then the PDF is $$f(z) = \begin{cases} z &amp; \text{for $0 &lt; z &lt; 1$} \\ 2-z &amp; \text{for $1 \le z &lt; 2$} \\ 0 &amp; \text{otherwise.} \end{cases}$$  How was this PDF obtained?   Thanks Calculate the limit including binomial coefficients Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true? Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true?    It seems to me like they are equal definitions in a way.    Can you give me a counter-example?    Thanks which of the two it is? How to embed Stack Overflow flair in a WordPress.com (free) blog? Has anyone successfully embedded their  on a WordPress.com-hosted blog?  The free hosted version of  (not )  from all posts before publishing. So they to strip out just about anything I can add to embed my Stack Overflow flair.  Note: I'll be switching to a paid, hosted site in the future (fixing the problem), but I'm looking for a quick, short-term solution. Early 80s anthology book about human kind being feared by other races in space I always thought the book was Manifest Destiny but its not. The story that sticks in my head was a human in a cage over a pit? Aliens walk in and discuss how dangerous humans were. He eventually escapes. Man I miss this book. wish I wouldve written down every book I ever read. Ugh I added a new field on Opportunity table to keep a count of records, as well as track the updates made on that record.  This is needed for an external system we use. Opportinity table already has around 5000 records now. I would like to update the new field that I added with some sequence order initially. Say each record to have a unique sequence number like 1,2,3 etc(can't use the Id field for unique). The field "Identifier__c" was added for this purpose with data type number and unique contraint. For that I created a temporary class and trigger. But it is throwing the error.     Review all error messages below to correct your data. Apex trigger OpportunitySAPIdentifierHandler caused an unexpected exception,   contact your administrator: OpportunitySAPIdentifierHandler: execution   of BeforeUpdate caused by: System.DmlException: Update failed. First   exception on row 0 with id 006E00000038FPSIA2; first error:   CANNOT_INSERT_UPDATE_ACTIVATE_ENTITY, OpportunitySAPIdentifierHandler:   execution of BeforeUpdate caused by: System.DmlException: Update   failed. First exception on row 0 with id 006E00000038FPTIA2; first   error: CANNOT_INSERT_UPDATE_ACTIVATE_ENTITY,   OpportunitySAPIdentifierHandler: execution of BeforeUpdate caused by:   System.DmlException: Update failed. First exception on row 0 with id   006E00000038FPSIA2; first error: SELF_REFERENCE_FROM_TRIGGER, Object   (id = 006E00000038FPS) is currently in trigger   OpportunitySAPIdentifierHandler, therefore it cannot recursively   update itself: [] Class.AutoNumber.OpportunityAutoNumber: line 76,   column 1 Trigger.OpportunitySAPIdentifierHandler: line 4, column 1: []   Class.AutoNumber.OpportunityAutoNumber: line 76, column 1   Trigger.OpportunitySAPIdentifierHandler: line 4, column 1: []:   Class.AutoNumber.OpportunityAutoNumber: line 76, column 1   Below is the trigger and class used.  trigger Opportuniwith sIdentifierHandler on Opportunity (before insert,before update) {          If(Trigger.isInsert || Trigger.isUpdate){             AutoNumber.OpportunityAutoNumber(Trigger.new);         }   }   The class is   public with sharing class AutoNumber {       public static void OpportunityAutoNumber (List&lt;Opportunity&gt; Opp) {          if(Trigger.isUpdate || Trigger.isInsert) {              for(Opportunity p: Opp) {                  List&lt;Opportunity&gt; oppList = new List&lt;Opportunity&gt;();                 oppList = [SELECT Id, createdDate, Identifier__c FROM Opportunity where Identifier__c = null and Id != :p.Id limit 100 FOR update];                 for (Opportunity oppL: oppList) {                      if(oppL.Identifier__c == null) {                          List&lt;Opportunity&gt; OpportunityList = new List&lt;Opportunity&gt;([select Identifier__c from Opportunity where Identifier__c != null order by Identifier__c desc limit 1]);                          if (!OpportunityList.isEmpty()) {                               Decimal maxval = OpportunityList.get(0).Identifier__c;                             oppL.Identifier__c = maxval + 1;                             update oppL;                         }                      }                     // integer currentCounter = Integer.valueOf(opp.Identifier__c);                     // oppr.Identifier__c = currentCounter + 1;                     // update oppr;                 }             }         }        }  } I'm currently working on the following question:  "At least one of Plato and Democritus believed in the theory of forms. Plato believed in the theory of forms only if he was not an atomist, and Democritus was an atomist only if he did not believe in the theory of forms. Democritus was an atomist. Prove, using Natural Deduction, that Plato was not an atomist"  I have been working for quite some time to take the information and turn it into a set of literals and statements I can work with.  So far, I have got $P \lor D$,  $P \Rightarrow \lnot a$,  $D \Rightarrow A$,  $D = A$  However, I cannot seem to make these prove the statement using natural deduction, any hints? Since the  would it make sense to create a Stack Exchange site dedicated to homework help? It wouldn't have to be limited strictly to homework either, possibly including discussions of puzzles and brain teasers and possible approaches to solving them.  I have a programming background so I would be interested in helping with CS or math related homework, discussions of programming contest problems, etc. Topics don't necessarily have to be related to programming either. That's just where my interests are.  What are your thoughts? I've had  on the web since 1999, and I'd like to add a page on why complex numbers are (or should be) considered to be numbers.  I'm frequently asked that question.  This is not a question about the existence or usefulness of complex numbers as they do exist as mathematical entities and they are very useful.  It is a question about what makes them numbers.  It could be broadened to ask what constitutes numbers, but I'm specifically interested in complex numbers.  Perhaps it's just historical and sociological, but I'm hoping for an answer that there's something intrinsic about complex numbers that makes them fit into the category of number. Why close and not migrate? Beamer itemize not aligned when in blocks While making a poster, I've got two columns with blocks with itemize inside them. MWE:  \documentclass[final]{beamer} \mode&lt;presentation&gt;{}  \usepackage{times} \usepackage{amsmath,amssymb} \usepackage[english]{babel} \usepackage[latin1]{inputenc}  \begin{document} \begin{frame}[t]{}   %\leavevmode \begin{columns}[t, onlytextwidth] \begin{column}{.5\linewidth} \begin{block}{Contributions} \begin{itemize} \item First Item \item Second Item \end{itemize} \end{block} \end{column} \begin{column}{.5\linewidth} \begin{block}{Approach} \begin{itemize} \item First Item \item Second Item \end{itemize} \end{block} \end{column} \end{columns}  \end{frame} \end{document}   My local installation and ShareLaTeX would misalign the bullets(baselines added by me):   This problem seems similar to:        But this time it happens when itemize is inside the block. If one removes the blocks it works fine.  Am I missing something or does it look like a Beamer bug?  Thanks! I'm a LaTeX noob, sorry if I missed something stupid. How to compute the following integral?  $$\int\log(\sin x)\,dx$$    Motivation: Since $\log(\sin x)'=\cot x$, the antiderivative $\int\log(\sin x)\,dx$ has the nice property $F''(x)=\cot x$. Can we find $F$ explicitly? Failing that, can we find the definite integral over one of intervals where $\log (\sin x)$ is defined? How should we understand the teletransportation thought experiment? Why the Principle of Least Action? How to remove uninstalled wine app icons from unity dash? I installed  (for Windows) using wine but turns out LAN Messenger is also available for linux as well. So..   I uninstalled LAN Messenger from wine. I also removed wine. deleted the .wine directory. after that sudo apt-get autoremove and sudo apt-get autoclean. then I installed LAN Messenger using dpkg -i pkgname.deb and it installed successfully.   The problem is that now I get this in dash:   Any Help is highly appreciated.  [NOTE] I am using ubuntu 12.10