Differential equation thought question Does anybody know how to solve the following differential equation?  $$w^2\cdot r+\frac{dw}{dt}\cdot r+\frac{dr}{dt}\cdot w=3g$$  where w and r are a function of time so $w(t)$ and $r(t)$. $g$ is a constant  This equation came from a thought problem of mine which I will now explain since I still do not know how to approach the problem.  Basically I want to design a spiral track in which a vehicle will accelerate. However the total acceleration needs to be constant.   The vehicle will have 2 acceleration components, a radial component ($a_r$) and a change in speed component ($a_s$). Since we want constant acceleration: $a_r + a_s = constant$  From circular motion we know that $a_r = \omega ^2 r$ where $r$ is the radius, and $\omega$    Further we know that $v=\omega*r$, therefore: $$a_s=\frac{d}{dt}(\omega r)$$  For constant acceleration $a_{total} = a_r + a_s = \omega ^2 r + \frac{d}{dt}(\omega r) =constant$  Further usefull information: Length of the track covered at a certain time $$l=v\cdot t=\omega r t$$  With all this the radius will be a function of time, $r(t)$ and the angular velocity will be a function of time, $\omega (t)$  Boundary conditions: $$r(0)  = 0$$  $$\omega (0) = 0$$  Now I would like to find a possible set of solutions to create a track by implementing end boundary conditions that i choose e.g. $v(t_{end})=100 [m/s]$ and $l(t_{end})=5000 [m]$  Here is a diagram for clarity:
First order differential equations with 2 varriables Does anybody know how to solve the following differntial equation?  $$w^2\cdot r+\frac{dw}{dt}\cdot r+\frac{dr}{dt}\cdot w=3g$$  where w and r are a function of time so $w(t)$ and $r(t)$. $g$ is a constant  EDIT: This equation came from a thought problem of mine which I will now explain since I still do not know how to approach the problem.  Basically I want to design a spiral track in which a vehicle will accelerate. However the total acceleration needs to be constant.   The vehicle will have 2 acceleration components, a radial component ($a_r$) and a change in speed component ($a_s$). Since we want constant acceleration: $a_r + a_s = constant$  From circular motion we know that $a_r = \omega ^2 r$ where $r$ is the radius, and $\omega$    Further we know that $v=\omega*r$, therefore: $a_s=\frac{d}{dt}(\omega r)$  For constant acceleration $a_{total} = a_r + a_s = \omega ^2 r + \frac{d}{dt}(\omega r) =constant$  Further usefull information: Length of the track covered at a certain time $l=v*t=\omega r t$  With all this the radius will be a function of time, $r(t)$ and the angular velocity will be a function of time, $\omega (t)$  Boundary conditions: $r(0)  = 0$ $\omega (0) = 0$  Now I would like to find a possible set of solutions to create a track by implementing end boundary conditions that i choose e.g. $v(t_{end})=100 [m/s]$ and $l(t_{end})=5000 [m]$  Here is a diagram for clarity:
Simple undivided plane UV map not behaving as expected Show reference label when it is undefined How to I deal with unwrapped UVs orientation?   Delete the default Cube. Add a Cylinder Edit mode and edge select the Top and the Bottom and a single connecting edge. Make them into seams UV Unwrap (standard Unwrap mode)   The unwrapped UV now shows two circles above a row of quads. I can save this and open it in an image editor and paint over those areas and get the texture to align on the cylinder back in blender.  But, the pattern(just happens to be text) that was inserted in place of the row of quads is now upside down on the model. Any attempt to flip or align just makes things worse, from upside down to mirrored or the Top/Bottom not aligning.  So my real question is how could I have known that the quads represented an upside down view of the world?  Do other blender users export UVs and mark them with arrows and colours just to see how they align on a model before actually painting for real?  I tried selecting the edges in step 3 in different orders before making them into seams and it made no difference.  I also looked at the Normals but they appear to be only Inside/Outside not Up/Down.  Blender 2.70 What is the maximum party size and the corresponding monster buffs? What is the maximum party size (if there is one) in Diablo 3?  Additionally, how much of a buff do monsters get when adding party members?     "A new ally has joined, but the minions of Hell grow stronger." What is an observer in quantum mechanics? My question is not about (pseudo) philosophical debate; it concerns mathematical operations and experimental facts.  What is an observer? What are the conditions required to be qualified of observer, both mathematically and experimentally? Does personified Death really exist in Harry Potter? I went through the last book, Harry Potter and the Deathly Hallows, yesterday and a question popped in my mind: Does Death, the supposed creator of the Deathly Hallows, really exist in the Harry Potter universe? Dumbledore thought otherwise and I am with the same school of thought.  By the way who created the deathly Hallows actually?  Update:  1) Why were the brothers so sure that the hooded figure was death(couldn't it be a more powerful wizard playing with them)  2) Couldn't, Beedle and the brothers create this story? Using divisibility theorems, prove that if $\gcd(a,b)=1$ and $a|c$ and $b|c$, then $ab|c$.   This is pretty clear by UPF, but I'm having some trouble proving it using divisibility theorems. I was looking at , and if this could be proven, then the solution would follow immediately. Parallel universes: are there an infinite amount of universes? Parallel universe: in every universe, you are presented with a different decision. Does that mean there are an infinite amount of parallel universes? I.e., if in one universe I choose to go left, I am then presented with different choices vs if i choose right, and that creates more universes with more decisions to be made.  Or does it mean that in the end of one's life we end up in the same place, meaning that its just one parallel universe that eventually collides with the other. A question occurred to me when browsing SE this evening, just curious.  What specifies the inside of a 3D construct? If I have a hollow sphere, what's to say that the world isn't the inside, and the center of the sphere isn't the outside. Picture the Asylum in the Hitchhiker's Guide to the Galaxy series, if you have read that.  Additionally, if the inside of a 3D object is defined as what is smaller than the outside, what happens when there is a a sphere with a volume equivalent to 51% of the universe? Web-mapping with editing capabilities? We know that in high temperature, resistivity in metals goes linearly with temperature. As temperature is lowered, resistivity goes first as $T^5$ due to "electron-phonon" interaction, and then goes as $T^2$ as temperature is further lowered due to "electron-electron" interactions.   My question is, is there an intuitive way of understanding the above power-law dependence at each temperature range physically? When to use RSA and when ElGamal asymmetric encryption Please explain this put option I don't see the statistics information that's supposed to be on the status bar running along the bottom of the main window towards the right in version 2.9. When I'm connected to two networks with separate routers using Wifi and Ethernet, how does my Mac know what connection to use when I visit a website? As stated , being on fire means you are doing well, and  we see that you can be commended as an MVP for being on fire for a large portion of the time. This  shows that destroying non-player things gives you fire points, but what all do I get points from, and how many fire points do I need to be on fire/stay on fire? Neural network references (textbooks, online courses) for beginners I want to learn Neural Networks.  I am a Computational Linguist. I know statistical machine learning approaches and can code in Python.   I am looking to start with its concepts, and know one or two popular models which may be useful from a Computational Linguistics perspective.  I browsed the web for reference and found a few books and materials.   Ripley, Brian D. (1996) Pattern Recognition and Neural Networks, Cambridge Bishop, C.M. (1995) Neural Networks for Pattern Recognition, Oxford: Oxford University Press. some links, like ,  (University of Toronto Psychology Department),  (University of Wisconsin Computer Science) and  (Facebook Research).   Coursera courses are generally nice, if anyone knows anything relevant from them. I prefer materials with lucid language and ample examples. edges with thickness and color Is there a way to vary the thickness and color of lines created by just edges and vertices? I realize that a face can be defined and then color and material applied, but is there any way to apply a color to the edges? Let $U$ be connected. A function $u \in H^1(U)$ is a weak solution   of Neumann's problem   \begin{equation} (*)\qquad\left\{ \begin{array}{rl} -\Delta = f &amp; \text{in } U \\ \frac{\partial u}{\partial \nu} = 0 &amp; \text{on } \partial U \end{array} \right. \end{equation}   if   $$ \int_U Du \cdot Dv \; dx = \int_Ufv \; dx $$   for all $v \in H^1(U)$. Let $f\in L^2(U)$. Prove $(*)$ has a weak solution   if and only if   $$ \int_U f \; dx =0. $$   For the only if part I set $v=1$. However I do not see where to start the if part. I was thinking of the Lax-Milgram theorem. That's where I'm now. I'm looking for an idiom that is equivalent to the Portuguese:  Quem tudo quer, tudo perde. Who wants everything, loses everything.  It basically points out that being too ambitious will have bad consequences: not only does one not get what one's striving for, but one also ends up losing what one already had (and probably took for granted). This idiom should be usable in everyday speech.