Converting a 2nd order O.D.E to two 1st order O.D.E's

[loinburger]

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...

[loinburger]

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!

[Jon Miller]

that is aneasy way

there are many other ways also



Jon Miller

[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.

[Jon Miller]

that was what I was expecting

Jon Miller

[Provost Harrison]

Sorry, did something boring wake me there?

[Hueij]

If you are suffering from O.D.E, shouldn't you go to a doctor ASAP?

[Urban Ranger]

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?

[loinburger]

Yup, the prof gave us a whole slew of nasty 2nd order O.D.E's to solve for the midterm (take-home test).

[Urban Ranger]

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.

[loinburger]

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.

[Urban Ranger]

Oh yeah, the programming is the hard part. C++? Fortran is the best for this, but you probably don't know it.

[loinburger]

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.)

[Urban Ranger]

Never used Mathematica before. IIRC, it looks sort of like Lisp.

[loinburger]

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.