Converting an IVP to a first order system I have a homework assignment in my Applied Numerical Methods class and I'm getting hung up on one of the questions.  The question is as follows:

I am really at a loss for how to do the first part of the question, where it asks me to convert the equation for a first order system.  I don't recall anything like this in class, so I'm not sure where to start.
PLEASE DON'T GIVE ME THE ANSWER.  I want to actually learn how do do this type of problem, so I'm really asking for the process rather than the result.
Can someone give me a hand at converting the IVP equation to a first order system and writing the discrete equations for the system?
 A: Typically, to convert to a first order system, one takes $x = u$ and $y = u'$, and then determines what equations $x$ and $y$ satisfy (i.e. $x' = \dots$, $y' =\dots$). 
For instance, take the $2$nd order ODE:
$$
u'' + u' + u = 1
$$
write $x = u$, $y = u'$, then $x' = y$ and $y' = u'' = - u' - u + 1 = -y - x + 1$
so your system is
$$
\begin{pmatrix}x \\ y \end{pmatrix}' = \begin{pmatrix}y \\ -x - y + 1\end{pmatrix}.
$$
In this case, the equation is linear, i.e. you can write
$$
\begin{pmatrix}x \\ y \end{pmatrix}' = \begin{pmatrix} 0 & 1 \\ -1 & -1\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix} + \begin{pmatrix}0 \\ 1\end{pmatrix}
$$
but this won't be possible in general (in particular, your equation is not linear, so don't be worried that you can't write it as a linear system)
A: To convert an $n$-th order  differential equation in one independent variable $x$ and one dependent variable $y$ to a system of $n$ first-order differential equations in $n+1$ variables $x_k$, you let
$$x_0 = x\\ x_1 = y \\ x_2 = y' \\ x_3 = y''$$and so forth. You can consider $x_0$ to be an independent variable.
Thus a second order equation becomes a "vector" of equations.  For example, for
$$ y'' + yy'x + x^2y^2 = x^3$$ you would have the system
$$ x_3 = x_2'\\
x_2 = x_1' \\
x_3 + x_1 x_2 + x_0^2 x_1^2 = x_0^3$$
