For example, given the differential equation $\dfrac{\mathrm{d}y}{\mathrm{d}x}=y\tag*{}$ We do, $\dfrac{\mathrm{d}y}{y}=\mathrm{d}x\tag{*}$ At which point we go and integrate both sides. Is there a rigorous justification for being able to go to $(*)?$
I see an equation like this:  $$y\frac{\textrm{d}y}{\textrm{d}x} = e^x$$  and solve it by "separating variables" like this:  $$y\textrm{d}y = e^x\textrm{d}x$$ $$\int y\textrm{d}y = \int e^x\textrm{d}x$$ $$y^2/2  = e^x + c$$  What am I doing when I solve an equation this way?  Because $\textrm{d}y/\textrm{d}x$ actually means  $$\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$  they are not really separate entities I can multiply around algebraically.    I can check the solution when I'm done this procedure, and I've never run into problems with it.  Nonetheless, what is the justification behind it?  What I thought of to do in this particular case is write  $$\int y \frac{\textrm{d}y}{\textrm{d}x}\textrm{d}x = \int e^x\textrm{d}x$$ $$\int \frac{\textrm{d}}{\textrm{d}x}(y^2/2)\textrm{d}x = e^x + c$$  then by the fundamental theorem of calculus  $$y^2/2 = e^x + c$$  Is this correct?  Will such a procedure work every time I can find a way to separate variables?
Is it possible to start windows with NUM Lock on? A password is always combination of characters &amp; numbers. I am so much used to num pad that every time windows start I end up entering wrong password just because NUM lock isn't ON by default. Since it's network login, it takes a bit long time to reset &amp; prompt for right password. Is there a way to enable NUM Lock on when Windows loads? How do I reset a lost administrative password? Infinite partition of $\mathbb N$ by infinite subsets Why is "idea" sometimes pronounced as "idear"? I know that idea is pronounced as /aɪˈdiə/, but I've meet several people in real life who put an 'r' at the end of the word. How come? What is the difference between a telephoto lens and a zoom lens? Looking at the , I notice a distinction between telephoto lens and zoom lens.  What is the difference between the two?  Why would I want one over the other?  I  but remain confused about this distinction.  Please talk to me like I'm stupid. :) I am hoping for a completely lay explanation.  In particular, I'd like to understand in what situations I'd use a telephoto lens, and in what (other) situations I'd use a zoom lens. Foobar2000-like music player? I have the following browsers on my phone:   stock Dolphin Mini Opera Mobile Firefox   I don't want to set a catch-all default because some pages render better on some browsers than on others.  When I click on a link on a non-browser app (e.g. google search or google reader) it asks me which browser to choose (as expected).  Can I set a default browser for some pages and a different default for others? 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? Why do rainbows have distinct colors? When I searched on the Internet for the reason of formation of , I got many explanations like  one &amp; . All the explanations consider only one spherical water droplet (like ).  Why don't the refracted waves coming from the adjacent droplets interfere with each other? For example, why doesn't a red ray coincide with a violet ray coming from another droplet at a suitable position?   How are we able to see rainbows with distinct colors? Since there are thousands of raindrops in the sky when it is raining, shouldn't it be a mess due to overlapping of different wavelengths?  Please explain. My laptop has 500gb hard disk in which for 250gb I installed Windows xp on C drive(primary partition). I want to install Ubuntu 12.04 in the remaing hard drive with dual boot option. When I am going to install Ubuntu it is showing something like  /boot /boot/   Can anybody help me in this regard that where and how to install Ubuntu? Proper Installation of Kali linux tools in Ubuntu Limit involving binomial coefficients: $\lim_{n\to\infty}\left(\binom{n}{0}\binom{n}{1}\dots\binom{n}{n}\right)^{\frac{1}{n(n+1)}}$ I am facing difficulty with the following limit. $$ \lim_{n\to\infty}\left(\binom{n}{0}\binom{n}{1}\dots\binom{n}{n}\right)^{\frac{1}{n(n+1)}} $$ I tried to take log both sides but I could not simplify the resulting expression. Please help in this regard. Thanks. sudo: unable to resolve host   I think I need to edit the /etc/hosts file to make an entry for 127.0.1.1 PC name.  But how can I do that if I can't access sudo? It seems to have gone since I updated to 13.10.  I have tried nano, which still doesn't work. i was stopped by my sister from upgrading Ubuntu, for the reason she explained was that i would have to reinstall all the softwares that I've installed in last 6 months. but i think if that's the case, the upgrade itself includes all softwares that i have installed. so please tell me what's the truth?     please note that i just can't afford mentally to reinstall all apps due to slow INTERNET speed.because it may take 3 to 4 days of constant downloading.   if you are an Indian living in an suburban city,you can understand my situation.   i think the question itself explains everything.still I'll explain everything in detail.i want to upgrade my current Ubuntu 12.10 OS to 13.10(now please don't ask what is 13.10). this option is available from "about this computer" and many other places.the upgrade is of about 700mb to 800 MB.    BTW, i don't know why i think it's a rule here to put some question on questions asked here,however detailed or/and simple it may be.at least that's what i feel after looking at many questions. The following problem was on a math competition that I participated in at my school about a month ago:      Prove that the equation $\cos(\sin x)=\sin(\cos x)$ has no real solutions.   I will outline my proof below.  I think it has some holes.  My approach to the problem was to say that the following equations must have real solution(s) if the above equation has solution(s):  $$ \cos^2(\sin x)=\sin^2(\cos x)\\ 1-\cos^2(\sin x)=1-\sin^2(\cos x)\\ \sin^2(\sin x)=\cos^2(\cos x)\\ \sin(\sin x)=\pm\cos(\cos x)\\ $$  I then proceeded to split into cases and use the identity $\cos t = \sin(\frac{\pi}{2} \pm t\pm y2\pi)$ to get   $$ \sin x=\frac{\pi}{2} \pm \cos x\pm y2\pi\\ $$  and the identity $-\cos t = \sin(-\frac{\pi}{2}\pm t\pm y2\pi) $ to get   $$ \sin x=- \frac{\pi}{2}\pm \cos x \pm y2\pi.\\ $$  where $y$ is any integer.  I argued that $y=0$ was the only value of $y$ that made any sense (since the values of sine and cosine remain between $-1$ and $1$).  Therefore, the above equations become   $$ \sin x=\frac{\pi}{2} \pm \cos x\implies \sin x \pm \cos x=\frac{\pi}{2}\\ $$  and   $$ \sin x=- \frac{\pi}{2}\pm \cos x\implies \cos x\pm\sin x= \frac{\pi}{2}.\\ $$  Then, by a short optimization argument, I showed that these last two equations have no real solutions.  First, does this proof make sense?  Second, if my proof makes sense, then I feel that it was not very elegant nor simple.  Is my approach the best, or is there a better (i.e., more elegant, shorter, simpler) proof? Add a "flag" button in the revision list How do I capture artificial light sources better? I am quite new to photography and was looking into how I can take better indoor photographs, particularly around night time.  I'm currently in a hotel room for the night and decided to try and take a picture of my room to test out my tripod which I haven't used before. However I noticed no matter what I set my camera settings to the artificial lights (particularly the lamp on the left, image attached) just appeared very washed out. You can't tell any detail about the lamp shade all you can see is just some sort of light but not much to it.  I tried a variety of fstop/shutter speed settings and tried a variety of ISO settings but they all appeared pretty much the same unless I massively underexposed the image.  Given that I wondered whether the solution would be to take a HDR image or take a normal exposure and an underexposed image and blend them later? I understand that artificial light like this can be very hard and harsh but I wondered if there's a way of improving the result with looking away from it?  I wasn't trying to take any particular special picture here but just test out my camera a bit more but this was the first thing I noticed and I didn't know how to fix it. Any tips would be greatly appreciated. How can I determine if a C++ object has been deallocated? If $f'(x)\le g'(x)$, prove $f(x)\le g(x)$ I have to do the following exercise:     Let $f$ and $g$ two differentiable functions such that $f(0)=g(0)$ and $f'(x)\leq g'(x)$ for all $x$ in $\mathbb{R}$. Prove that $f(x)\leq g(x)$ for any $x\geq0$.   Now, I know this is true because the first derivative of a function is the angular coefficient of the function in a point $x$. So, $f'(x)\leq g'(x)$ means, in other words, that the function $g(x)$ grows faster than $f(x)$. I think this is the base for a more formal proof, could someone help me to figure out a more formal proof? Align equation left Here is my split equation:  \documentclass{article} \usepackage{amsmath} \begin{document} \begin{equation} \begin{split} a_{ijk} =&amp; \frac {Pr(M_{I} =2  \&amp;  M_J=1 \&amp; M_K =1 | I=i , J=j , K=k)}{Pr (M_I =1 \&amp; M_J =1 \&amp; M_K=1 | I=i , J=j)} \\             =&amp; \frac {\mu_{ijk211}}{\mu_{ijk111}}\\ \end{split} \end{equation} \end{document}   I want to align this equation to the left. Is this possible in an equation environment?