Stationary points of an ODE-system (again)

There is another exercise where I am given a non-linear system of ODEs and a question that confuses me:

Consider $$x'=x(y+1)\\ \text{ and }\\ y'=xy+2.$$

"Determine stationary points of the system, and linearize the system at the stationary points, determine the geometric type of the linearizations and the stability."

The only stationary point is $(x=2,y=-1)$. For the stability, I wrote $$f(x)=x(y+1),\\ g(y)=xy+2,$$ then differentiated: $$f'(x)=y+1,\\ g'(y)=x.$$ At $(-1,2)$, these values are $0$ and $2$. For $f$, I wrote that we cannot use linear terms in the expansion and hence would have to consider $f''$ and so on. For $g$, since it's greater than zero, the point is not linearly stable in $y$-direction.

But what is meant by "determine the geometric type of the linearizations"? What are types of linearizations? What am I missing here?

all the best,

Marie

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2022-07-25 20:46:33
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