ODEs of order larger than 1

An ODE of order n can be written in the form

y⁽ⁿ⁾(t) = f(t, y, y',..., y^(n-1))

and has initial conditions

First order systems of ODEs

A first order system of ODEs can be written in the form

y₁' = g₁(t, y₁, ..., y_n) , ..., y_n' = g_n(t, y₁, ..., y_n)

or, equivalently,

y' = g(t, y)

where y, y', g(t, y) are column vectors.

Converting higher oder ODEs to first order systems

You have to do this to solve the ODE numerically with Matlab. Let y₁=y, y₂=y', ..., y_n=y⁽ⁿ⁾. Then the above n-th order IVP becomes the first order system

y₁' = y₂
...
y_n-1' = y_n
y_n' = f(t, y₁, y₂,..., y_n)

with the initial conditions