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Math Help - System of two second order diff equations

  1. #1
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    System of two second order diff equations

    Some ideas?

    A6*x + A4*(y')^2 - 2*A2*x'' - A3*x*y'' + A4*y*y'' = 0
    A5 - A3*(x')^2 - A3*x*x'' + A4*y*x'' - 2*A1*y'' = 0

    '=(d/dt), ''=(d^2/dt^2), Ai-known constants. The initial conditions are:

    x(0)=a, y(0)=0, x'(0)=0, y'(0)=0.

    Thank you in advance!!!

    Last edited by Ackbeet; April 4th 2011 at 03:14 AM. Reason: Get it down to an unreplied thread. Better for the OP.
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  2. #2
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    So is that a system of coupled ODE's in the independent variable time? That is, if dots represent differentiation with respect to time, then is your system

    A_{6}\,x+A_{4}\,\dot{y}^{2}-2A_{2}\,\ddot{x}-A_{3}\,x\ddot{y}+A_{4}\,y\ddot{y}=0

    A_{5}-A_{3}\,\dot{x}^{2}-A_{3}\,x\ddot{x}+A_{4}\,y\ddot{x}-2A_{1}\,\ddot{y}=0?

    If so, that's an extraordinarily nasty system of ODE's. Probably only solvable numerically.
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  3. #3
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    Yes, you are right Mr Ackbeet. I spent much time to find analytical solution, but nothing. This describe a phisicaly problem whery I use some aproximation to make a system simplier. Otherwise, the system is non-linear and non solvable analytical like this simplier system as you say. Can I get a numerical solutions for initial conditions (x(0)=0,y(0)=a,x'(0)=0,y'(0)=0) but in function of constants Ai and t. I want to see how phisicaly model depend in time but not for concretly numerical values of Ai? Thank you for take this into consideration. Very helpful suggestions dear professor Ackbeet.
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  4. #4
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    If you're interested in how the solutions depend on the constants A_{i}, there might be some tricks you could play. I remember doing something like this in my Ph.D. dissertation, in Mathematica. You have to use the := "delayed definition" operator in Mathematica. Check out the code in Appendix A. I describe that code on pages 26ff. There I'm trying to show how the complex eigenvalue \xi depends on the parameter k.

    Warning: the code in Appendix A took two hours to run on a typical machine at the time of writing. It would probably still take an hour or so even now, and there's not much guarantee it would even run in the newer versions of Mathematica. But, the code might give you some ideas you could run with.

    Part of your issue is that you're interested in so many different parameters. Technically, though, you could normalize both your equations so that one particular coefficient is just 1, and then re-label your parameters. It wouldn't change your solution at all. So then you'd be down to just four parameters.
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  5. #5
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    I apply a Mathematica 8 for many problems. I can see that the problem is very serious. Yes, but I can reduce a number of parameters on 3 or 4, I suppose. I will see your suggestion. If you can send me some example of code for one or two parameters in case for some simple diff equation. I will try to solve in Mathematica for my case. Anyway, if I could reduce a number of parameters, I will post the simplier system how directly depend of material constants (Some constants are repeated through the Ai, and I will make a case where we will have 3 constants, Ai[a,b,c]). Thank you again. You make a very good and applicable thing with good suggestions. I will understand a important physical problem if we do something together with this diff equations.
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  6. #6
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    derdack,

    I'd love to be able to do that for you, but I think it would take me more time than I'm willing to put into it. You can get the code from my thesis, along with the explanation as I mentioned above. For specific Mathematica commands, just google or look them up in Mathematica's help file.

    The problem as you describe it is a bit higher level than MHF is typically used for. You might try Math Overflow for more in-depth help.
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  7. #7
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    You are right. Thank you Mr Ackbeet!
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  8. #8
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    You're welcome!
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