$f$ and $g$ are continuously differentiable functions such that $f(x) = g'(x)$ and $g(x) = f'(x)$ and any product of $f, g, f', g'$ is commutative for all $x \in \mathbb R$  I need to show that $f^2 - g^2 = C$ , where $C$ is a constant.  Logically this makes sense if you for example use a trig function $\sinh(x)$ but how can I show this to be true?
Suppose $f$ and $g$ are continuously differentiable functions such that $f(x) = g'(x)$ and $g(x) = f'(x)$ and that any product of $f, f', g$ and $g'$ is commutative for all $x ∈ R$. Show that $ f^2 − g^2 = C $ for some real constant C  I have actually no clue how to solve this, and would be really greatfull for all the help i can get
Right-click on Touchpad not working after update to 18.04 LTS Is it a bad practice to catch Throwable? ANOVA assumption normality/normal distribution of residuals Setting environment variables on OS X What is the proper way to modify environment variables like PATH in OS&nbsp;X?  I've looked on Google a little bit and found three different files to edit:   /etc/paths ~/.profile ~/.tcshrc   I don't even have some of these files, and I'm pretty sure that .tcshrc is wrong, since OS&nbsp;X uses bash now. Where are these variables, especially PATH, defined?  I'm running  (Leopard). What is another word for 'all round' capabilities? I am looking for a positive, passionate word to describe the all-round capabilities of a person.   Are there any words better than 'all-rounder'? How to Implement Page Object and Page Factory Pattern in Selenium Webdriver? I am a beginner in learning selenium Webdriver. I am not able to understand how page object and factory models works and how it can be beneficial?    Kindly help me explain it with detailed explanation. Ways to update Ubuntu (debian-based distros) offline What is the easiest way to update/upgrade an offline Ubuntu machine with the help of a Windows machine that has internet access?  This question also applies to installing new packages offline and not only updates/upgrades.  The Linux system shall remain completely cut off from the Windows machine, meaning there should be no ethernet connection or the likes between the two computers.  Update:  Please have patience, since I haven't updated this since two weeks. There are some complications getting apt-offline to work properly on Lubuntu and all other methods seem to be too old and deprecated:  AptOff - last update 2006 AptMedium - last update 2007 Keryx - last update 2011, looks like this project was discontinued  I hope I will be able to somehow update the offline machine, finally.  While I was still not able to update/upgrade the offline system due to authentication errors, I accepted an answer, because it's taking too long. Explaining Horizontal Shifting and Scaling I have tried as following:  1. Turn off: export DISPLAY=:0.0 &amp;&amp; xset dpms force off  2. Turn on: export DISPLAY=:0.0 &amp;&amp; xset -dpms   Which works but that is a temporary solution, which does not fix my problem yet. When i reboot system it start again and again the same auto turn off mode.   How can i now remotely using SSH turn it off forever?  IMPORTANT:  - KIOSK (ATM machine will use this option) - 24/7 advertising display/slides will use this   Optional Note: this guide i followed   , but permanent solution did not worked. Can we model gravitation as a repulsive force? This question is actually related to my earlier question ("").  The fact that objects move a lot in the universe and that the universe is expanding, can imply that gravity is a repulsive force that increases with distance.. so the farthest objects repel us more.  This can still explain several existing observations, e.g., why does the apple fall?   Motion is the result of such repulsion. Two objects unlucky enough not to be moving relative to each other get squished due to the repulsion of the rest of the universe  around them. The earth repels the apple less than the stars so it is pushed towards the earth.   Furthermore, it can explain the expanding universe without the need for dark energy.  This could be demonstrated in a thought experiment. If we take a lot of same-charge particles (with small mass) such as electrons and lock them in a large box at a low enough temperature. The mutual repulsion of the particles may cause similar motion as if due to gravitational attraction.   Another experiment would be to measure the slight changes in our weight during day and night when the sun and earth align (if their masses are large enough to detect the feeble change in repulsion).   [EDIT: the question in the original form may not have been clear. It is "can we model".. with a yes/no answer and why (not). If downvoting, please justify. Securing SSH server against bruteforcing I have a little SVN server, old dell optiplex running debian. I don't have that high demands on my server, because its just a little SVN server... but do want it to be secure.  I just renewed my server to a newer and better optiplex, and started looking a bit into the old server. I took it down after experiencing problems. When I check the logs, its full of brute-force attempts and somehow someone has succeeded to enter my machine. This person created some extra volume called "knarkgosse" with two dirs "root" and "swap1" or something.  Don't really know why and what they do, but sure do want to prevent this from happening again. I find this a bit strange though because I change my password ever few months or so, and the passwords are always random letters and numbers put together... not easy to brute-force.  I know I can prevent root from logging in, and use sudoers... and change the SSH port, but what more can I do?  So I have a few questions:   How can I prevent logging in for 5 minutes after X amount of incorrect tries. Or slow tries down after each incorrect try? Is there some kind of central blacklist which a server can connect to? A blacklist that keeps track of IP addresses that are "unsafe" and should never be granted access? What more can I do to apply safety to my server?   Like I said earlier, I am running Debian 5 with Apache (www-data user problem?), svn, mysql, php, phpmyadmin, hudson. It is on a home network with port forwarding on 80, 443, 8080, 8180, 23 and 22. Why do games run slow on CPU Maybe my question sounds naive because I'm not a game developer but I dont see objective reasons for this. An average cpu is in times more powerful than an average gpu like a few graphics cards in one. The most repeated reasons in internet:  the GPU isn't as busy as the CPU  If the gpu renders the screen how it can not be busy The modern cpus are powerful enough. They are not busy most of time. For example my cpu's usage is 3%.  many GPU's are designed to process graphics commands in parallel, CPU single threads them  What bothers the games to run a few parallel processes even on one cpu core? In my opinion its not correct to tell that the gpu is better if games never use 100% of the cpu's capacity. To be specific they dont use even 50%.  To clarify my question based on responses in the comments:  Did you mean to ask &quot;Why do games render more slowly on the CPU?&quot;  Yes I mean why the cpu cant do the whole work for games when it is underloaded and especially if it does the largest part of work in any case.  You can not compare GPUs and CPUs directly, because they do different things.  What bothers them to do the same things?  The CPU isn't as well suited to performing rendering calculations  I dont understand what &quot;suited&quot; means. And why the cpu is not suitable for primitive tasks aka paint roller.  adding additional parallel processes doesn't improve performance  Then why do you produce multicore graphics cards and multicore central processing units? Just give various parts of one task to a few processes if one process is slow and 100% of the capacity is not used. What bothers to emulate the gpu? Its also unclear for me why when you play without a separate graphics module the games see only one cpu core when you play with the gpu they see and use all the cpu cores. May be some magic. $\mathbb{R}$ \ $\mathbb{Q}$ and $\mathbb{R}^2\setminus\mathbb{Q}^2$ disconnected? ContourPlot of implicit function doesn't work when I square the equation I'd like to create a picture like the one with TikZ:    I've already created the electrical part using circuitikz. I'd like to add the part on the right (the motor's shaft with arrows), but don't know how to start, actually. Is there a way to draw 3D shapes, maybe? Or should I draw circles and rotate them?  \documentclass{article} \usepackage{tikz} \usetikzlibrary{intersections} \usepackage{circuitikz} \usepackage{siunitx}  \begin{document}  \begin{figure} \centering \begin{circuitikz} \draw (0,0) to [V, v=$v(t)$, *-*] (0,2)      to [R, l=$R_d$] (2,2)     to [L, l=$L_d$] (4,2)     to [V, v&lt;=$e(t)$] (4,0) to [generic] (4,2) -- (4,0) -- (0,0); \end{circuitikz} \end{figure}  \end{document}   What I have now:   What I'm still missing: Sometimes when I am debugging code in Eclipse it happens that although I can see and inspect class member variables without any difficulty I am unable to inspect the values of variables declared locally within functions. As an aside, any parameters to the current function lose their 'real' names and instead one sees their values listed in the Variables window as arg0, arg1, arg2, etc but at least the values are visible.  This is occurring at present in relation to classes defined within the core JDK. I have verified that the installed and current JRE is a JDK.  Is anybody able to shed some light on this behaviour? Why must a planet be in a star's "habitable zone" to have life? Backup of clients' directory to Windows Server 2008 We have a machine running Windows Server 2008, and we need to take backup of folder from a desktop on LAN. This folder is not shared.   Is there any tool or software to do this, or can a scheduled task be made which should automatically take the backup of that folder date-wise &amp; make a copy on server? Ie can we do it through Schedule Tasks option ?  Desktop systems have Windows Vista/Windows 7. How do I add new wallpapers? I have Ubuntu 11.10 installed and want to download and install additional wallpapers on my desktop.   How do I do this? In definition of vector multiplication, the direction of the resulting vector is given by the right hand rule. However I don't know any mathematical requirement to pick right hand instead of left for this purpose. Is there any?