Reducing a 2nd order system of ODEs to a 1st order system I need to numerically solve the following system of ODEs:
$$x''(t)=- \frac{3x}{(x^2+y^2)^{3/2}}$$
$$y''(t)=- \frac{3y}{(x^2+y^2)^{3/2}}$$
I know that I must convert to a system of 1st order equations but I am having trouble doing so...Could someone provide me with a starting point? This problem is called the "one-body problem" apparently but I cannot find much about it as all the hits I'm getting involve the 2 body problem...Help please.

All that needs to be done to convert to a system of first order ODEs is to play some games with your notation, in this case it's particularly simple because you don't have any first derivatives.
So let's say we choose to define
\begin{align}
x_1 &\equiv x, \\
x_2 &\equiv x_1', \\
y_1 &\equiv y, \\
y_2 &\equiv y_1'.
\end{align}
So now your system system will consist of 4, first order, coupled ODEs in your 4 variables. It will look like the following
\begin{align}
x_1' &= x_2, \\
x_2' &= \frac{-3x_1}{\left( x_1^2 + y_1^2 \right)^{3/2}}, \\
y_1' &= ..., \\
y_2' &= ... .
\end{align}
Are you now able to write down the final two equations in your system?
Now you can just numerically solve these 4 ODEs for each of your 4 variables; $x_1, x_2, y_1, y_2$. You can then relate these back to your original $x, y$ through their respective definitions.
