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 Posted: Wed Sep 23, 2009 6:59 pm
Hi. I have been somewhat discouraged by my lack of understanding of differential equations... which I realize is pretty important for physics. I think part of my problem is that I didn't work very hard in the first class I had in it (linear algebra and ordinary differential equations) and later on when I wanted to learn partial diff eqs I couldn't get satisfactory answers for the general questions I had because it was expected I already knew it all. Maybe I should have tried to take ODEs again but there is so much physics I have to fit into my schedule already. Anyway, I was wondering if some of you here who really know this can give me a few suggestions?
I understand how specific differential equations will satisfy physical conditions but I want to understand the mathematics in a more general way (kind of the point, I guess), but in a way so I still can understand what the problem is actually talking about.
First, what are the solutions for some of the types seen often in physics? Like 1st order, 2nd order linear... and is what is the basic method used for obtaining them? When you find a homogeneous solution and you want to get a particular solution from it what should you do? And also, does the homogeneous solution have any special importance physically? How are eigenfunctions used? sweatdrop
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Posted: Wed Sep 23, 2009 7:52 pm
Others can probably answer better, but I'm currently in my second semester of Differential Equations, so I can answer most of your basic questions.
1st order linear can be solved in a few ways, based on their type. The types include:
1st Order Linear
1st Order Seperable
1st Order Exact
Each of them has a technique for solving that type specifically. You can solve 1st Order more generally, but these are a shortcut if you can recognize the type.
To solve nth order linear ODEs, you need to find a particular solution, and a fundamental set of solutions, and the general solution is the sum of these.
You mentioned specifically finding particular solutions. Note that particular solutions are only needed in non-homogeneous Linear ODEs. Where the confusion may arise is that to solve a get the fundamental set from a non-homogeneous, you treat it like it's homogeneous. The particular solution requires you to use the non-homogeneous version, however.
To actually solve for a particular solution, there are a few options. The first is the Method of Undetermined coefficients, which my prof so lovingly refers to as "Educated Guessing". Basically, you make a guess at what approximately the solution should look like, and then solve for the necessary coefficients that would make it true. However, unless your ODE is a fairly simple one, this method is not recommended. It is very easy to have a guess that is ever so slightly off, and then all your work is wasted as you need to start over again.
Cons: Fairly easy to have a bad model
Pros: Much much faster and easier than Variation of Parameters if your model is good.
The other method is Variation of Parameters, which is very computational. Hence why the first method is advised for most simple ODEs. And if you think you've got a good guess, it may still be beneficial to attempt Undetermined Coefficients before you use this method. There are plenty of Wronskians and integrals in this one.
Cons: Computationally Dense
Pros: Guaranteed to give you an answer
Eigenfunctions, eigenvalues, and eigenvectors are all used to solve systems of linear ODEs. Explaining how they work exactly isn't my best area of Differential Equations, but needless to say they're used to solve for your 2 unknown functions. sweatdrop
If you'd like to know about any particular methods, feel free to ask. (It would've been a monumental task to answer everything in detail in one post.)
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Posted: Thu Sep 24, 2009 4:19 pm
Mecill I understand how specific differential equations will satisfy physical conditions but I want to understand the mathematics in a more general way (kind of the point, I guess), but in a way so I still can understand what the problem is actually talking about. I don't know of a good ODE book that was specifically written for physical intuition, but one I have a lot of respect for is Simmons, George F., Differential Equations with Applications and Historical Notesit spends quite a bit of time on scientific applications of what it's talking about, and as the name implies, it is also liberally sprinkled with (typically, but not always, brief) biographical and historical data on where equations or the techniques for solving them come from, which I often found quite interesting in itself. On the other hand, if you just want quick experience for cheap, I'm sure there's a Schaum's book for ODEs. Depending on your time constraints, that might be better. Mecill First, what are the solutions for some of the types seen often in physics? It is probably no exagerration that the most important differential equation in physics is the second-order linear ODE, and the most important form of it is the Sturm-Liouville equation. Straightforward examples are the Bessel, Legendre, Hermite, Laguerre, Chebyshev, and Airy equations, as well as a host of hypergeometric special functions; a generalization (inhomogeneous) comes up quite a bit as well. Frequently, the process of solving PDEs gives a SL system as a subproblem. A general method involves Green's functions. Mecill When you find a homogeneous solution and you want to get a particular solution from it what should you do? That depends on the relationship between the homogeneous and inhomogeneous equations; if you simply dropped the inhomogeneous term, try something else. On the other hand, if you explicitly homogenized Py = f (P a differential operator) into APy = 0 (A another differential operator that annihilates f, i.e., Af = 0), then finishing the problem from the homogenized solutions to the latter is basically equivalent to the method of undetermined coefficients sans the actual guessing: the homogenized solutions tell you what to work with explicitly. Mecill And also, does the homogeneous solution have any special importance physically? Mathematically, they form a vector space, opening the door to some powerful results. Practically, you need them to for the general solution, and hence typically essential in fitting some initial or boundary conditions. Mecill How are eigenfunctions used? By either Deep Math or Furious Handwaving, the general solution is in many cases going to be a linear combination of eigenfunctions. I couldn't even begin a general enough treatise here, so I'll give a particularly simple example. Say you have a PDE ∂f/∂t = Af, where A is a linear operator independent of t, the natural thing to do is try finding separable solutions f(t,x) = F(x)G(t), so: F(x) dG/dt = G(t) [AF(x)] 1/G(t) dG/dt = 1/F(x) [AF(x)] which is a curious equality, because both sides in fact be constant for it to hold. Call this constant λ. Then the problem is reduced to finding solutions to AF = λF, which is precisely the eigenvalue problem of linear algebra. The most trivial example of an eigenfunction expansion is the Fourier series. Geometrically, the terms of the Fourier series are projections of the function to the Fourier basis {sin nx, cos nx} (and it's not a trivial fact that this is, in fact, a basis in, say, the a suitable vector space of functions). Things of that sort arise very naturally in boundary value problems; even a simple harmonic oscillator y" + λy = 0, when subject to conditions y(0) = y(a) = 0, is going to have solutions only for particular values of λ, the eigenvalues of the problem, an a general solution is going to be a linear combination of the corresponding eigenfunctions. I'd rather you ask a more specific question.
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Posted: Thu Sep 24, 2009 4:50 pm
Hi! Thanks very much for replying, Chaotic Nonsense! smile
Is the general way that you mentioned to solve first order equations the method you would use for nth order with n=1?
I looked up 1st order exact: http://mathworld.wolfram.com/ExactFirst-OrderOrdinaryDifferentialEquation.html
p(x,y)dx + q(x,y)dy = 0
D_yp = D_xq
I'm not sure I understand the part about the requirement for the conservative field. Maybe... what would be an example in physics of a first order exact equation?
So, the integrating factor method is for the inexact 1st Order ODEs. Is this a different general method or is it also related to the nth order general method?
=>I'm still not quite clear on what is probably an important point: The general solution is the sum of the particular solution and the fundamental set of solutions, or is it just the sum of the fundamental set is the general solution? And what is the reasoning behind this? Why is the general solution "general?"
I don't have much background in solving on computers but I'm curious if possible to use a program to do variation of parameters? I have a hard time with the idea of guessing a solution and then finding coefficients to make it fit but I guess that method is used frequently so maybe I have to accept it. lol.
Maybe I should attempt to make a chart to keep track of some of the categories of equations and methods... It would be easier than having to flip pages. Has anyone ever seen anything like that printed?
--
Thanks for your reply too, VorpalNeko! I will try to ask more specific questions about what you wrote tomorrow but now I have to go work on homework. sweatdrop
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Posted: Thu Sep 24, 2009 6:27 pm
Mecill I looked up 1st order exact: http://mathworld.wolfram.com/ExactFirst-OrderOrdinaryDifferentialEquation.html p(x,y)dx + q(x,y)dy = 0 D_yp = D_xq I'm not sure I understand the part about the requirement for the conservative field. For a first-order ODE of the form p dx + q dy = 0, where p,q are functions of x,y only, consider F = [p;q] a vector field. Then [1] ∂p/∂x + ∂q/∂y = 0 iff F is solenoidal [2] ∂q/∂x - ∂p/∂y = 0 iff F is irrotational If this is not clear, imagine adding one more dimension with a constant component, and computing ∇· F and ∇✕ F; in the latter case, pay attention to the third component of the curl, since the first two trivially vanish. Since the first part of [2] is just the statement that the ODE is exact, F is irrotational, and hence can be written as a gradient of some potential. Those are equivalent characteristics of what it means for a vector field to be "conservative"; physically, a conservative force field has a work integral that's path-independent, which can be found by the gradient theorem: -W = Int[ ∇f·d s ] = f( b) - f( a), where a and b are the endpoints of path of integration, and we can unambiguously say that conservative forces are those that have a scalar potential as a function of position. I mention the other condition because solenoidal fields are those that can be written as a curl of a vector potential (example: magnetic field, ∇· B = 0 implies B = ∇✕ A). Interesting side note: for complex z = x+iy, [1] and [2] together are equivalent to F(z) = p - qi being holomorphic. Mecill Maybe... what would be an example in physics of a first order exact equation? : find the shape of a mirror such the reflected rays of light from a point-source are parallel. (Which may be quite an issue if you're designing a flashlight or a satellite dish.) Thermodynamics: It's absolutely essential. Look up "exact differential" and "Maxwell relations"; you'll see that the two-variable exact differential has the the same requirement as the first-order exact ODE. So any suitable process which keeps that exact differential zero involves an exact first-order differential equation. Mecill So, the integrating factor method is for the inexact 1st Order ODEs. Is this a different general method or is it also related to the nth order general method? Integrating factors aren't going to go away. They're very important to order reduction, and some other things. For example, the aforementioned Sturm-Liouville equation can be found from any linear second-order ODE by a suitable integrating factor. Mecill =>I'm still not quite clear on what is probably an important point: The general solution is the sum of the particular solution and the fundamental set of solutions, or is it just the sum of the fundamental set is the general solution? And what is the reasoning behind this? Why is the general solution "general?" General = Particular + Homogeneous. The reason, of course, is that one can add a homogeneous solution to any particular solution to get another solution. This works in a parallel way to the "+C" from ordinary integration--one can think of Int[ f(x) dx ] = F(x) + C as just another way of stating a differential equation y' = f, with the F being a particular solution and C being a homogeneous solution (dy/dx = 0). As point of terminology, the "fundamental set" is simply the set of all homogeneous solutions. Mecill I don't have much background in solving on computers but I'm curious if possible to use a program to do variation of parameters? I'm not sure, but MAPLE might have a tool that would walk you through it. I don't know; I typically use a computer to either do it for me completely or to check my final answer.
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Posted: Thu Sep 24, 2009 8:01 pm
You're most certainly welcome. I consider VorpalNeko and Layra-chan to be the authorities on mathematics here, so if you can make sense of what they're explaining -not always easy with the text limitations- definitely go with it. biggrin
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Posted: Fri Sep 25, 2009 6:36 pm
VorpalNeko It is probably no exagerration that the most important differential equation in physics is the second-order linear ODE, and the most important form of it is the Sturm-Liouville equation. Straightforward examples are the Bessel, Legendre, Hermite, Laguerre, Chebyshev, and Airy equations, as well as a host of hypergeometric special functions; a generalization (inhomogeneous) comes up quite a bit as well. Frequently, the process of solving PDEs gives a SL system as a subproblem. A general method involves Green's functions. OK. I'm going to post the ones you mentioned so we can look at the actual equations:  BTW is there any program I can get for free that will allow me to type up equations easily? I have seen some greens functions before but did not understand how it was a general method for solving DEs. Also, hypergeometric, meaning geometric in more than 3 dimensions? And SL system is a sturm-liouville system? VorpalNeko Mecill When you find a homogeneous solution and you want to get a particular solution from it what should you do? That depends on the relationship between the homogeneous and inhomogeneous equations; if you simply dropped the inhomogeneous term, try something else. On the other hand, if you explicitly homogenized Py = f (P a differential operator) into APy = 0 (A another differential operator that annihilates f, i.e., Af = 0), then finishing the problem from the homogenized solutions to the latter is basically equivalent to the method of undetermined coefficients sans the actual guessing: the homogenized solutions tell you what to work with explicitly. So... Should I never just drop the inhomogenous term? I'm not really sure what you mean about the way of homogenization with the operators. How do you get the operator AP? VorpalNeko Mecill And also, does the homogeneous solution have any special importance physically? Mathematically, they form a vector space, opening the door to some powerful results. Practically, you need them to for the general solution, and hence typically essential in fitting some initial or boundary conditions. Oh! This sounds cool. What are some of the results from having them form a vector space? Thanks very much! smile Your replies on the other stuff were also helpful. Also, I have the book Simmons and Krantz - Differential Equations: Theory, Technique, and Practice. Do you know if it is similar to the one with historical notes? I have not read it much, though I probably should try again sometime. I could also get the one with historical notes from the library...
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Posted: Fri Feb 12, 2010 4:26 pm
Hi, Mecill! I'm coming into this thread kinda late, so I apologize ahead of time for either covering something someone else has already said or confusing you more. But my main questions are: What level of college are you in and what maths/physics classes have you had so far? What maths/physics do you want to use differential equations to do? My impression is that while Vorpal's responses are wonderfully accurate, they're a bit higher-level/off-topic than you were shooting for. Mecill BTW is there any program I can get for free that will allow me to type up equations easily? Do you have experience with TeX/LaTeX? If you're doing physics, you'll probably need to use it at some point. Layra and Vorpal have some way of getting TeX output into their Gaia posts, but I was never diligent enough to follow their advice on how to do it sweatdrop As for the functions that Vorpal mentioned...they are important to physics but are also pretty advanced. The most important thing to note is that they're all linear, second-order differential equations. Second-order equations come up a ton in physics because of the basic reasoning of physics: if you have the initial position and velocity (2 initial conditions) then you should have a unique solution. This can only happen, in most cases, with a second-order equation. The equations are usually linear or semilinear because the velocity and acceleration might depend on position and time, but usually not directly on velocity. (A good counterexample to this is air resistance, where the force of friction increases proportional to velocity.) Quote: I have seen some greens functions before but did not understand how it was a general method for solving DEs. Also, hypergeometric, meaning geometric in more than 3 dimensions? And SL system is a sturm-liouville system? In reverse order: SL does stand for Sturm-Liouville. Hypergeometric functions are a class of functions that are power series where the coefficients in the expansion a_n are rational functions of n. I have no idea why they're called hypergeometric, except possibly because they're linked to elliptic integrals. These integrals are called elliptic because the first example of them gives the arc length of an ellipse. Green's functions are a pretty general method for solving inhomogeneous DEs with boundary conditions, if you can actually find the function. The basic idea is that you want to solve an equation like Lu = f for u, where L is a differential operator and f is a known function, with boundary conditions. If you can find the Green's function, which is specific to the operator L and the boundary conditions, you can just do an integral and solve that equation. This makes Green's functions incredibly powerful; the caveat is that they're often hard to find and plenty of problems simply don't have one. Quote: When you find a homogeneous solution and you want to get a particular solution from it what should you do? ... So... Should I never just drop the inhomogenous term? I'm not really sure what you mean about the way of homogenization with the operators. How do you get the operator AP? Uhhh...I can't really follow what's going on here. I think you guys are just missing each other. Can you give an example of a particular inhomogeneous equation you've come across, to demonstrate this idea? Quote: VorpalNeko Mecill And also, does the homogeneous solution have any special importance physically? Mathematically, they form a vector space, opening the door to some powerful results. Practically, you need them to for the general solution, and hence typically essential in fitting some initial or boundary conditions. Oh! This sounds cool. What are some of the results from having them form a vector space? As a physicist, the primary thing you should associate with the term "vector space" is "superposition principle". Anytime you have a linear homogeneous equation, the solutions are going to form a vector space. This means that if you add solutions, multiply them by constants - everything you can do in a normal vector space - you get another solution. Waves are the prototypical examples for this, and superposition is extraordinarily important for electrodynamics and quantum mechanics (light waves and wavefunctions). So, anytime you have a system like this, you know you're gonna get an infinite number of solutions and you're gonna know how to generate new solutions from old ones (adding and multiplying). Another concept that comes out of vector spaces is spectral theory. Take normal 3-dimensional vector space; you know there are an infinite number of vectors, but you can express any of them in terms of the three basis vectors. Same thing here: complex solutions can (usually) be given as simple expressions involving basis functions. If you ever take a class on Fourier analysis or quantum mechanics, you'll get this technique shoved down your throat (mostly because it's insanely practical and useful!). The last reason to have them form a vector space, as Vorpal said, is for getting particular functions out of the boundary conditions. Say you have a standing wave on a string. You can solve the wave equation and get an infinite number of solutions. But now suppose you know the starting position and velocity of the string. This is a physical situation, so if its really a deterministic system, it only has one solution. Now you need to use the spectral technique to pick that one solution out of the infinite possibilities. You would need to do this for any other system as well, but it's much, much, much harder if the solutions don't form a vector space. Damn, I was trying to keep this post short. Hope it helps a little.
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Posted: Fri Feb 12, 2010 5:55 pm
Hey, Swordmaster Dragon. It's nice to meet you.
Since I posted this thread last fall I have actually learned quite a bit more about differential equations but I know there is still a lot I haven't payed enough attention to, so perhaps revisiting this topic will be helpful. I'm now a senior undergrad. I want to go to grad school eventually for some kind of physics, but I'm not sure what yet, and I'm not going to grad school right next fall.
I have a little experience with TeX. I have it installed on my computer but I haven't worked with it much since I installed it.
About the thing with the operators: I didn't understand how you would make Py = f into APy = 0 at first because I didn't understand how A would annihilate f, but I think I understand it now... What I was asking was how do you find A for an equation, and it depends on the equation, right?
So, now that I'm studying Quantum Theory more seriously some of the equations Vorpal mentioned are relevant to that. Yeah, I need to study spectral theory more. (I actually didn't know that was called spectral theory.)
I checked out another book on mathematical physics (Butkov) from the library that goes through some of this so I'm going to read it.
Thanks very much for your explanations of Green's functions and the importance of vector spaces. They made sense to me quite well! smile
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Posted: Sat Feb 13, 2010 6:14 pm
Mecill Since I posted this thread last fall I have actually learned quite a bit more about differential equations but I know there is still a lot I haven't payed enough attention to, so perhaps revisiting this topic will be helpful. I'm now a senior undergrad. I want to go to grad school eventually for some kind of physics, but I'm not sure what yet, and I'm not going to grad school right next fall. I see. That's kewl, I'm taking a year off before grad school, too. Have you thought about what you're going to do in that time? Quote: I have a little experience with TeX. I have it installed on my computer but I haven't worked with it much since I installed it. If you've done any programming (even just HTML) TeX should come pretty easy. Do a few tutorials for a couple weekends and I'm sure you'll pick it up quick. Still, I don't know how to get TeX into Gaia XD Quote: About the thing with the operators: I didn't understand how you would make Py = f into APy = 0 at first because I didn't understand how A would annihilate f, but I think I understand it now... What I was asking was how do you find A for an equation, and it depends on the equation, right? It depends on the function f. Think: if Py = f and Af = 0, then (AP)y = A(Py) = A(f) = Af = 0 so APy = 0. You need to find an A that annihilates f. The finesse comes with figuring out which A to pick; a lot of operators are going to annihilate f, but only a few are going to make the problem APy = 0 easier. Quote: So, now that I'm studying Quantum Theory more seriously some of the equations Vorpal mentioned are relevant to that. Yeah, I need to study spectral theory more. (I actually didn't know that was called spectral theory.) Okay, yeah. I didn't know if you had done quantum mechanics yet. If you hadn't, Vorpal's equations wouldn't be very helpful. But now that you're in the thick of it, a lot of them should look familiar. Spectral theory is a pretty general and damn useful concept. It basically means any situation where you can express a large number of complicated things (e.g. entire vector space) in terms of a small number of simpler things (e.g. the eigenvectors of an operator) - the spectrum of the complicated stuff. Most important for physics are 1) using normal modes of coupled oscillations to describe the general motion and 2) using eigenfunctions in QM to figure out the time evolution of a general wavefunction. Quote: I checked out another book on mathematical physics (Butkov) from the library that goes through some of this so I'm going to read it. I haven't found a good explanatory math. phys. book yet, so definitely tell me if you like that one. I've read through Arfken and Weber (Mathematical Methods for Physicists) which is very thorough but much more of a reference than a textbook. Quote: Thanks very much for your explanations of Green's functions and the importance of vector spaces. They made sense to me quite well! smile I'm glad! I put a lot of weight on teaching and have spent a lot of time studying it. Anytime you don't understand something I say or have a suggestion for communicating things, don't hesitate to tell me :3
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Posted: Thu Feb 18, 2010 6:02 am
Swordmaster Dragon I'm taking a year off before grad school, too. Have you thought about what you're going to do in that time? No, I have no idea what I'm going to do. *sigh* I tend to avoid long term planning, but hopefully I'll find something interesting that will help me be better prepared for grad school. XD Swordmaster Dragon If you've done any programming (even just HTML) TeX should come pretty easy. Do a few tutorials for a couple weekends and I'm sure you'll pick it up quick. Still, I don't know how to get TeX into Gaia XD Ok, tutorials sound like a good idea! Thanks for the encouragement. Probably to get it into Gaia would be the same as on any forum with bbcode? Swordmaster Dragon Spectral theory is a pretty general and damn useful concept. It basically means any situation where you can express a large number of complicated things (e.g. entire vector space) in terms of a small number of simpler things (e.g. the eigenvectors of an operator) - the spectrum of the complicated stuff. Most important for physics are 1) using normal modes of coupled oscillations to describe the general motion and 2) using eigenfunctions in QM to figure out the time evolution of a general wavefunction. Ah, okay! I'm working on getting this stuff down. For some reason, though the concept of orthogonality makes sense to me I have difficulty using it in practice. I think think it's because the first class I had that it explained it was a difficult experience so I blocked it out.... Swordmaster Dragon I haven't found a good explanatory math. phys. book yet, so definitely tell me if you like that one. I've read through Arfken and Weber (Mathematical Methods for Physicists) which is very thorough but much more of a reference than a textbook. Yeah, I actually own Arfken and Weber. I agree that it's a great reference book, so I could probably learn a lot from just studying that alone, but some things I just wanted to see a different sort of explanation of so... This one is written much differently. To me it reads more like a straight math book. It says it's intended for self-study by theoretical students. I'm sure the time you spent on learning ways to teach and study effectively will serve you well in grad school. smile
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Posted: Sun Feb 21, 2010 7:09 am
Mecill Ok, tutorials sound like a good idea! Thanks for the encouragement. Probably to get it into Gaia would be the same as on any forum with bbcode? Yeah...I've never really been on any other forum. Maybe when I get my other computer up and running again I'll try to look it up. In fact, a mini-tutorial by Layra or Vorpal or any of a number of other people might be buried in the archives here. Quote: Yeah, I actually own Arfken and Weber. I agree that it's a great reference book, so I could probably learn a lot from just studying that alone, but some things I just wanted to see a different sort of explanation of so... This one is written much differently. To me it reads more like a straight math book. It says it's intended for self-study by theoretical students. Thinking back to reading through it, while Arfken/Weber was really focused on math, it had a *lot* of practical examples from physics applications. If the book you have turns out to be too abstract, I would flip through that, see some of the examples, and if that doesn't help go find a physics book that the examples are from. I didn't understand a lot of practical differential equations methods until I could relate it back to DE's I had already solved in electromagnetism, classical mechanics, and quantum mechanics. Speaking of DE's, I'm about to post a problem (that I'm having a problem with) in the main forum if you want some practice with a not-so-simple DE xp
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Posted: Sun Mar 07, 2010 6:01 am
Swordmaster Dragon Mecill Ok, tutorials sound like a good idea! Thanks for the encouragement. Probably to get it into Gaia would be the same as on any forum with bbcode? Yeah...I've never really been on any other forum. Maybe when I get my other computer up and running again I'll try to look it up. In fact, a mini-tutorial by Layra or Vorpal or any of a number of other people might be buried in the archives here. I'll look in the archives someday, I guess... The other forum I sometimes visit is PhysicsForums, btw. Quote: Speaking of DE's, I'm about to post a problem (that I'm having a problem with) in the main forum if you want some practice with a not-so-simple DE xp Cool,Thanks! Yeah, I've looked at the post. I'm just been taking some time off from Gaia for a while, sorry for leaving in the middle of our convo...
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