Take the conservative equation $x'' + x^2 - 9 = 0$.

Write down the corresponding first order system using the extra variable $y$:
$x' = {}$
$y' = {}$

The critical points are on the x-axis at the following two x values (order them as they are on the number line):


Compute the Jacobian matrix for all $x$ and $y$: $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$

The behavior at the first critical point is
The behavior at the second critical point is
For behavior write one of: "saddle", "source", "sink", "spiral sink", "spiral source", "center".