Let $X(t) = \left[\begin{matrix}a(t) &amp; b(t) \\c(t) &amp; d(t)\end{matrix}\right]$ and let $U$ be a nonsingular matrix. How do you solve $$\frac{d}{dt} X(t)=UX(t)$$ I presume there is some general method to solve these kinds of ODEs but I cannot find anything about it, online.   BTW: -----------------  I know that you can get 4 ODE equations for the four unknown functions of $t$. The problem is that each ODE equation includes other functions as so $$\frac{d a}{d t}=u_{11}a+u_{12}c$$
Solve the following system of equations  $ \begin{cases}  x_1^{'}(t)=x_1(t)+3x_2(t) \\  x_2^{'}(t)=3x_1(t)-2x_2(t)-x_3(t) \\  x_3^{'}=-x_2(t)+x_3(t)\end{cases} $.  First, I create the column vectors $X$ and $X^{'}$. Then the matrix $$A= \begin{bmatrix} 1 &amp; 3 &amp; 0  \\ 3 &amp; -2 &amp; -1 \\ 0 &amp; -1 &amp; 1  \\ \end{bmatrix} $$  Now, I find the eigenvalues, $-4,3,1$ and their corresponding eigenvectors $(-3,5,1)^T (-3,-2,1)^T (1,0,3)^T$.   I'm just not sure how to take it from here and solve the system of differential equations. I want a diagonal matrix $D$ so that I can read the solutions easy, but I'm not sure how to do it.  EDIT   Building on @Francisco 's answer, I'd have that:  $$X=c_1  (-3,5,1)^T  e^{-4t}   + c_2 (-3,-2,1)^T e^{3t} + c_3 (1,0,3)^T e^{t} $$.  But I believe this could be written in a simpler form.
How do I find out what fonts are used in a document/picture? Finding what's using all the space in *nix Which sequence of commands will tell me which files are the largest starting from a particular directory, including all sub directories? I want to know where all the space went.  Preferably just with regular unix'y commands.  If possible, try to keep it compatible with Sun OS 5.10 (perhaps in addition to regular Linux answers, not as a replacement, to keep it as general as possible.) Sass .scss: Nesting and multiple classes? Which B2C option to choose? A stock Salesforce org is geared toward B2B companies, but I work in Higher Ed and we're B2C-focused.  There seem to be three main B2C options available:   Person Accounts Nonprofit Starter Pack (NPSP) 1-to-1 model NPSP bucket model   Why would I choose one option over the other? Rolle's Theorem and the Mean Value Theorem Let $f\colon \mathbb{R} \to \mathbb{R}$ have derivatives of all orders. Suppose that $a &lt; b$ and that $f(a)=f(b)=f'(a)=f'(b) =0$. Prove that $f'''(c) = 0$ for some $c$ in $(a,b)$. Regular expression: match start or whitespace Can a regular expression match whitespace or the start of a string?  I'm trying to replace currency the abbreviation GBP with a £ symbol. I could just match anything starting GBP, but I'd like to be a bit more conservative, and look for certain delimiters around it.  &gt;&gt;&gt; import re &gt;&gt;&gt; text = u'GBP 5 Off when you spend GBP75.00'  &gt;&gt;&gt; re.sub(ur'GBP([\W\d])', ur'£\g&lt;1&gt;', text) # matches GBP with any prefix u'\xa3 5 Off when you spend \xa375.00'  &gt;&gt;&gt; re.sub(ur'^GBP([\W\d])', ur'£\g&lt;1&gt;', text) # matches at start only u'\xa3 5 Off when you spend GBP75.00'  &gt;&gt;&gt; re.sub(ur'(\W)GBP([\W\d])', ur'\g&lt;1&gt;£\g&lt;2&gt;', text) # matches whitespace prefix only u'GBP 5 Off when you spend \xa375.00'   Can I do both of the latter examples at the same time? Graphics horribly corrupted on Ubuntu 16.04 in VirtualBox, showing random strips of pixelated colors Inside a VirtualBox VM, I have installed myself an Ubuntu 16.04.1 LTS Server installation. However, when I installed the ubuntu-desktop package and rebooted, this happened:    Exact command I used: sudo apt-get update; sudo apt-get install ubuntu-desktop ..and then a reboot command.  This just seems like a corrupted top-left portion of the screen, as my cursor can appear in view (but corrupted):    This is a purely graphical issue because I can use the keyboard to successfully progress to a desktop:    Note: On startup it shows correct graphics for a split-second then reverts to glitch.    Any help on this? When I go to Settings &gt; Applications &gt; Manage Applications &gt; Downloaded I see an application list. If I select some applications I see button "Force stop" active, but some applications have this button dimmed.  What does it mean? What happens if I press this button?  This button can't mean application running, since the applications with this button are not listed in Running section.  Can I list just applications which have the Force stop button active and uninstall them?    Android version: 3.2 Regular cast vs. static_cast vs. dynamic_cast How can I check which songs or videos that I purchased from the iTunes Store have DRM restrictions?  All of the mobile OSes are very impressive to me at the moment, and I'm thinking about trying a non-iOS device again. I want to know now the impact it has on my Apple-sourced music collection though.  Is there a date cut-off for purchases when items went DRM free? How to make partitions mount at startup? How do you make HDDs and partitions mount at startup on Ubuntu?  I always keep my media and data files on separate partitions - one 2tb HDD and a 400 odd gig partition.  I have been trying to swap the default folder location for the home directory to my other drives.  It works when I change the /home/user_name/.config/user-dirs.dirs file to the directories I want, but only until I reboot the system. I am only assuming that the drives not mounting is the problem with it, but it would be helpful either way.  If any one knows another reason for why the directories change back to default each time that would be good also. GAP: how to work with elements of the group ring of the symmetric group $\mathbb{C}[S_k]$? How to show that $\lim_{n \to +\infty} n^{\frac{1}{n}} = 1$? I've spent the better part of this day trying to show from first principles that this sequence tends to 1. Could anyone give me an idea of how I can approach this problem?  $$ \lim_{n \to +\infty} n^{\frac{1}{n}} $$ Questions about insects eating flour and living in the cracks of a wooden cutting boards I have a large wooden cutting board which is used in my kitchen mainly for preparing egg pasta. The wood type is Populus (poplar or aspen or cottonwood).  The main ingredients used with the cutting board are: flour (mainly soft wheat flour), hens' eggs, white sugar, butter, and baking powder.  After the food preparation, the cutting board is cleaned by scraping away the ingredients remains.  The board has some cracks and some small unknown insects are living inside the cracks.   The board has also a lot of holes due to woodboring beetles but I think the woodworms operated long time ago and they no longer live in the cutting board.  The unknown insect has an approximate length of 1 mm (along the direction of their movements) and a width of 0.3 mm. The unknown insects do not have wings, they just walk. I am not able to see how many pairs of legs they have.  The color of the insect is pale brown.  The cutting board was free from these unknown insects some months ago but then they appeared.  My questions:   What insects are they? Are these insects poisonous to humans? How can I get rid of them?   Update  The unknown insects could be Acarus siro, I will try to get a picture of the insects.  Update 2  Here is a picture. The bottom part of the image is a one Euro coin, the top part of the image is a caliper (each bright area is one mm tick).   Some more pictures: Wrong ordering in generated table in jpa Keeping track of programs When I install a simple program it often uses make &amp;&amp; make install and doesn't often even have an uninstall target.  If I wish to upgrade a program, is it standard protocol to assume it just rewrites seamlessly over the old program?  How do I keep track of these programs; do most people just 'fire and forget' and if no uninstall target is given do I have to manually delete everything? Can I use variables inside {} expansion without `eval`? Can I use variables inside {} expansion without appealing to eval? If so, how?  This doesn't work:  $ touch 1.foo 1.bar $ ls 1.{foo,bar} 1.bar  1.foo $ extensions=foo,bar $ ls 1.{$extensions} ls: cannot access 1.{foo,bar}: No such file or directory   It works with eval:  $ eval ls 1.{$extensions} 1.bar  1.foo Merge duplicate records into 1 field Karn, the Great Creator - what defines a 'card from outside the game' in sealed? In the upcoming set 'War of the Spark' there is the Planeswalker . His -2 ability reads as:      -2: You may choose an artifact card you own from outside the game or in exile, reveal that card and add it to your hand    Now my question is, in a sealed tournament, like the Prerelease, is a player allowed to also choose cards from outside his sealed cardpool, or only out of the cards therein. Does anyone know any rulings on stuff like that? How can I make an object follow a path exactly