Is there a way to derive the second linearly independent solution to ODE whose characteristic quadratic has double roots?
Where does $xe^x$ solution come from when the characteristic polynomial is square?
Kinetic energy is simple, right?! from the equation it seems as though if you know the speed and mass you'll know the work. Let's try to prove it. let's assume we know the acceleration and distance for now. $$w=mda$$ So the distance is $$d= 1/2 *at^2$$ Then  $$t = \sqrt{2d/a}$$ So $a*t$, which is the speed, is  $$a*\sqrt{2d/a} = a* \sqrt{2d}/\sqrt{a} = \sqrt{2da} = \sqrt{2w/m}$$ So regardless of the distance or acceleration, if you know the work and the mass, you can calculate the speed which is  $$v=\sqrt{2w/m}$$  What if you know the speed and the mass? You can easily calculate the work. $$v = \sqrt{2w/m}$$ Then  $$v^2 = 2w/m$$  That would mean  $$m*1/2*v^2=W $$ Rearrange it and  $$W= 1/2 *m *v^2$$ would be the equation.  What if you have an initial speed and a new speed and want calculate the amount of work done after an initial speed to get the new speed? You'll just subtract the total work from the initial speeds work.  $$1/2mv^2 - 1/2ms^2$$  where s is  the initial speed and v is the new speed. However if you do it the other way which is  $$d= 1/2at^2 + vt$$ and solve it the answer would be $$1/2ms^2  + 1/2msv$$ Why aren't they equal? What reputation do you need on Stack Overflow to receive an invite to Careers? How to reduce vertical space in matrix? I want to reduce the size of the vertical space in matrix environment and have a result something like this     But at present I manage something like below     with the following code    $\begin{bmatrix}     00 &amp; 08 &amp; 10 &amp; 18 &amp; 20 &amp; 28 &amp; 30 &amp; 38 \\     01 &amp; 09 &amp; 11 &amp; 19 &amp; 21 &amp; 29 &amp; 31 &amp; 39 \\     02 &amp; 0a &amp; 12 &amp; 1a &amp; 22 &amp; 2a &amp; 32 &amp; 3a \\     03 &amp; 0b &amp; 13 &amp; 1b &amp; 23 &amp; 2b &amp; 33 &amp; 3b \\     04 &amp; 0c &amp; 14 &amp; 1c &amp; 24 &amp; 2c &amp; 34 &amp; 3c \\     05 &amp; 0d &amp; 15 &amp; 1d &amp; 25 &amp; 2d &amp; 35 &amp; 3d \\     06 &amp; 0e &amp; 16 &amp; 1e &amp; 26 &amp; 2e &amp; 36 &amp; 3e \\     07 &amp; 0f &amp; 17 &amp; 1f &amp; 27 &amp; 2f &amp; 37 &amp; 3f   \end{bmatrix}$   I can change the font using \mathsf{}, but other than that, how I reduce the vertical space between the rows? I did find things like  to increase vertical space between rows in matrix, but not something that can decrease the space. I recently got a Windows 10 laptop. Opposite to my Macbook, I must swipe right-to-left to go to the right virtual desktop. Is there a way to reverse this, so I swipe left-to-right to go to the right virtual desktop? What does the "Current event" post notice mean? The recently introduced &quot;Current event&quot;  feels incomplete:  At the very least, it needs a period at the end to complete the sentence. Aside from that, what is a future visitor of a question supposed to take away from this notice? The &quot;citation needed&quot; text explains what expectations are placed on the question bearing the notice and invites action. Is the &quot;current event&quot; notice purely informational? What's the desired effect of adding it to a question? The current on-question notice is:  This post relates to a rapidly changing event.  In addition to answering those questions, I'm hoping we can work out a proposal for a wording change to make it more clear. I can't come up with anything to start with, since I don't quite get what the notice is supposed to achieve in the first place. I have a table with a column for ID, Date and Frequency of occurrence for that particular date and ID.  ID    date          Frequency 1     2014-05-18      5 1     2014-05-19      4 1     2014-05-20      25 2     2014-05-20      7 3     2014-05-18      4 3     2014-05-20      1 4     2014-05-18      6   I'm looking to insert the dates for each ID that aren't currently there (assuming that the date ranges from 2014-05-18 to 2014-05-20) and add a value of 0 in Frequency for those.  For example, you can see that  ID 1 has an observation for all three dates;  ID 2 is missing 2014-05-18 and 2014-05-19;   ID 3 is missing 2014-05-19;   ID 4 is missing 2014-05-19 and 2014-05-20.  Therefore, the output should look like this  ID    date          Frequency 1     2014-05-18      5 1     2014-05-19      4 1     2014-05-20      25 2     2014-05-18      0 2     2014-05-19      0 2     2014-05-20      7 3     2014-05-18      4 3     2014-05-19      0 3     2014-05-20      1 4     2014-05-18      6 4     2014-05-19      0 4     2014-05-20      0   I've attempted the following from . However, that will only add dates with 0 Frequency that aren't present throughout the entire table instead of on an ID level.   DECLARE @s DATE = '20140518', @e DATE = '20140520';  SELECT c.date, Frequency = COALESCE(a.Frequency,0) FROM dbo.Calendar AS c LEFT OUTER JOIN dbo.table AS a ON c.date = s.[date] WHERE c.d &gt;= @s AND c.d &lt; DATEADD(DAY, 1, @e); What is the reason for the observation that across the board fields in physics are generally governed by second order (partial) differential equations?    If someone on the street would flat out ask me that question, then I'd probably mumble something about physicists wanting to be able to use the Lagrangian approach. And to allow a positive rotation and translation invariant energy term, which allows for local propagation, you need something like $-\phi\Delta\phi$.  I assume the answer goes in this direction, but I can't really justify why more complex terms in the Lagrangian are not allowed or why higher orders are a physical problem. Even if these require more initial data, I don't see the a priori problem.  Furthermore you could come up with quantities in the spirit of $F\wedge F$ and $F \wedge *F$ and okay yes... maybe any made up scalar just doesn't describe physics or misses valuable symmetries. On there other hand in the whole renormalization business, they seem to be allowed to use lots and lots of terms in their Lagrangians. And if I understand correctly, supersymmetry theory is basically a method of introducing new Lagrangian densities too.   Do we know the limit for making up these objects? What is the fundamental justification for order two? What did Dumbledore say his scar on his left knee was? In Philosopher's Stone, we hear Dumbledore replying to McGonagall's query as to whether he could do something about the scar baby Harry has on his forehead:     “Even if I could, I wouldn’t. Scars can come in handy. I have one   myself above my left knee that is a perfect map of the London Underground." - Chapter 1, The Boy Who Lived.   Now, I have no idea what that last sentence means.  It could be because English is not my native tongue. Is it some kind of a joke, or does it really mean something? I want to initialize a struct element, split in declaration and initialization. This is what I have:  typedef struct MY_TYPE {   bool flag;   short int value;   double stuff; } MY_TYPE;  void function(void) {   MY_TYPE a;   ...   a = { true, 15, 0.123 } }   Is this the way to declare and initialize a local variable of MY_TYPE in accordance with C programming language standards (C89, C90, C99, C11, etc.)? Or is there anything better or at least working?  Update I ended up having a static initialization element where I set every subelement according to my needs. Given a message and DES encrypted form of said message, is it possible to efficiently compute the key used to encrypt the data? Unsealed meat in freezer Below is my situation:  Is completely (vacuum) sealing leftover meat in the freezer absolutely necessary? I would always just use a plastic bag to cover the meat and fold it around couple of times so it's almost as good as sealed (usually grocery plastic bag is much bigger than the piece of meat) and have never had a problem with it last a decade or so I have been cooking.  But recently I got this new roommate who insists on me using special small plastic bags with sealers to use and he is being very paranoid, obsessive about it.  So my question is: Is he right? compare two columns of different files and print if it matches I am using Solaris 10 and so grep options involving -f don't work.  I have two pipe-separated files:   file1:  abc|123|BNY|apple| cab|234|cyx|orange| def|kumar|pki|bird|   file 2:   abc|123| kumar|pki| cab|234   I would like to compare the first two columns of file2 with file1 (search through the entire contents of file1 in first two columns) if they match print the matched line of file1. Then search for the second line of file 2 and so on.   Expected Output:  abc|123|BNY|apple| cab|234|cyx|orange|   The files I have are huge, containing about 400,000 lines, so I would like to make the execution fast. If $Y$ is connected, why is $A\cup Y$ connected in this case? Word for "not consciously aware of" What's the word to convey the meaning "not consciously aware of"? Assume that $f$ be continuous on $\mathbb{R}$, $f'(x)$ exists for all $x\neq 0$, and $\lim_{x\rightarrow 0} f'(x)=1$. We need to show $f'(0)$ exist and is equal to $1$.  $f'(0)=\lim_{x\rightarrow 0}\frac{f(x)-f(0)}{x}$, $\lim_{x\rightarrow 0}f'(x)=1\Rightarrow\lim_{x\rightarrow 0}\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=1$...am I going in the right direction? Please help. Allow the owner of a post to approve a suggested edit even if it was rejected by reviewers I'm looking for a library/method to parse an html file with more html specific features than generic xml parsing libraries. What should be created first in a video game? i am finding nowhere a solution I am running UBUNTU 12.04 and using GNOME desktop, which I installed additionally. Logging into classic GNOME I see the following:     WHAT is this "earth globe" icon coming from?Seems to be named "starter". It is configured to open a website, from my bookmarks. WHY is it not showing the TIME?   Using dconf-editor I had no luck in answering the above questions.  Any idea? Finding the limit of roots products $(\sqrt{2}-\sqrt[3]{2})(\sqrt{2}-\sqrt[4]{2})(\sqrt{2}-\sqrt[5]{2})\cdot \cdot \cdot (\sqrt{2}-\sqrt[n]{2})$