15-02-2003 22:10									#1
	Converting a 2nd order O.D.E to two 1st order O.D.E's Support Apolyton buy from Amazon

Is there a general formula that can be used to convert a 2nd order O.D.E into a system of two 1st order O.D.E's (or an Nth order O.D.E into a system of N 1st order O.D.E's)? F'rinstance, I'd like to convert the O.D.E.
y" - (1 - y²)y' + y = 0, [y(0) = .5, y'(0) = .5]
into two 1st order O.D.E's
u' = f(u, y)
y' = f(y, t)
...but I've no idea where to begin. I can solve 1st order O.D.E's without too much difficulty, but 2nd order O.D.E's are hating on me...

	15-02-2003 22:36									#2
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Whoops, never mind, I managed to find my old diff eq textbook hiding under a pile of junk in the basement.
y' = u
u' = (1 - y²)u - y
[y(0) = .5, u(0) = .5]
Woot! Take that, second order!

	15-02-2003 22:41									#3
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that is aneasy way

there are many other ways also



Jon Miller

	15-02-2003 22:47									#4
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I'll probably only need the easy way. I'm numerically solving these things, so the 1st order O.D.E's don't need to look especially pretty -- they just need to be 1st order.

	15-02-2003 23:28									#5
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that was what I was expecting

Jon Miller

	15-02-2003 23:32									#6
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Sorry, did something boring wake me there?

	15-02-2003 23:41									#7
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If you are suffering from O.D.E, shouldn't you go to a doctor ASAP?

	16-02-2003 10:14									#8
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quote: Originally posted by loinburger I'll probably only need the easy way. I'm numerically solving these things, so the 1st order O.D.E's don't need to look especially pretty -- they just need to be 1st order.

Oh. For numerical analysis?

	16-02-2003 10:22									#9
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Yup, the prof gave us a whole slew of nasty 2nd order O.D.E's to solve for the midterm (take-home test).

	16-02-2003 10:28									#10
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Do you need to write your own program, or use something like Maple or Mathematica?

I have always hated take home exams, they are always evil beyond belief.

	16-02-2003 10:31									#11
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We need to submit our own, but we're allowed to test our solutions against Mathematica or Matlab in order to get error estimates, and we're also allowed to borrow code (our text is chock full of pseudocode).

Yeah, basically take-home exams are an excuse to give a 20-hour test. Fortunately, once I get my method programmed in, the rest should be pretty easy.

	16-02-2003 10:34									#12
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Oh yeah, the programming is the hard part. C++? Fortran is the best for this, but you probably don't know it.

	16-02-2003 10:38									#13
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Mathematica or Matlab. I don't know Matlab very well, but I've programmed O.D.E. solvers in Mathematica before. (Those were just basic implementations of Euler's method, though, while these equations are probably going to need something more like an Adams-Multivalue or Bulirsch-Stoer, but the principle's the same.)

	16-02-2003 11:07									#14
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Never used Mathematica before. IIRC, it looks sort of like Lisp.

	16-02-2003 19:48									#15
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Yeah, it's basically LISP with better built-in functions (it autmatically handles derivatives, f'rinstance). It's also not so anal about the parentheses.

	16-02-2003 20:27									#16
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Hurm. I much rather use Fortan than Lisp. Lisp is probably the worst language to do numerical analysis in, even though it's awesome doing symbolic stuff.

	18-02-2003 01:24									#17
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Hey UR, do you have any idea how to perform Richardson extrapolation using stepsizes h and h/i (with i > 1)? All the material I've found on it only employs stepsizes h and h/2, which works fine most of the time, but I've got an unbelievably stiff diff eq and I don't want to wind up computing a thousand points for my tableau...

	18-02-2003 10:53									#18
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Let me think ( ) about it. The bad thing about postgrad work is you are supposed to do a bit of original work.

	18-02-2003 10:58									#19
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Bah humbug.