Introductory Programming
Fall 2005

Evaluation 7 on Tuesday


Second order systems
--------------------

The nice people in Natick have given us a tool that can
estimate the solution to any first order diff EQ; that
is, anything that can be expressed as

y' = f(t, y)

All we have to do is write a function that implements f

It takes t and y as input variables and produces the
value of y' at t and y as a result.


So what do we do with second order systems?

y'' = f(t, y, y')

ode45 can't handle them like that.

One solution is to replace y  with r
                      and  y' with v

Then we can write

a = f(t, y, y')

dv/dt = y'' = a

dr/dt = v


And then define a vector [r, v], now

dX/dt = [dr/dt, dv/dt] = [v, a]

And ode45 can handle that fine.

We just have to write a function that takes t and X,
and computes dX/dt.

Everywhere there used to be a scalar y, we use a vector X.

Punchline: you can write an nth-order diff EQ as a system of
	   n first-order diff EQs