Find a differential equation, not involving time and solve to find a implicit equation for this dynamical system

Question

Question

I am concerned with part c) of this question.

In part b I found there to be 5 critical points at

$$(0,0) , \quad (0,-2) , \quad (2,0) , \quad (2,-2) , \quad (1,-1)$$

And linearizing around these point I find

At $(0,0)$: $\pmatrix{-2 & 0 \\ 0 & -2}$,

at $(0,-2)$: $\pmatrix{2 & 0 \\ 0 & 2}$

at $(2,0)$: $\pmatrix{2 & 0 \\ 0 & 2}$

at $(2,-2)$: $\pmatrix{-2 & 0 \\ 0 & -2}$

at $(1,-1)$: $\pmatrix{0 & -1 \\ -1 & 0}$

And from this I found that the crit points $(2,0)$ and $(0,-2)$ are unstable

$(0,0)$ and $(2,-2)$ are stable

and $(1,-1)$ is a saddle

Now for part c) I really don't know what it is asking me to do

Does it want me to find an equation for what $x$ and $y$ are equal to in the linearized system?

Any hints as to what the question means would be very helpful

Answer 1

A hint for c):

If a prime denotes differentiation with respect to $x$ one has $$y'={\dot y\over\dot x}\ .$$