# Linearization of nonlinear system and the behavior of the linearized system

Consider the nonlinear system:

\begin{cases} x' = x^2 + y \\\\ y' = x - y + a \\\\ \end{cases} where $a$ is a parameter.

$a)$ Find all equilibrium points and compute the linearized equation at each.

For this question I solve \begin{cases} x^2 + y = 0 \\\\ x - y + a = 0 \\\\ \end{cases} to give me $\displaystyle x = \frac{-1 \pm \sqrt{1 - 4a}}{2}$ and $\displaystyle y = \frac{-1 \pm \sqrt{1 - 4a}}{2} + a$ as equilibrium points. The linearized system I got is \begin{cases} x' = y \\\\ y' = x - y \\\\ \end{cases}

$b)$ Describe the behavior of the linearized system at each equilibrium point?

Can someone help me with this one? The linearized system doesn't depend on $a$ and so why does it ask for the behavior at each equilibrium point?

The linearization does depend on $a$. Your linearization is missing a term. Setting up the Jacobian, you get: \begin{bmatrix} 2x & 1\\ 1 & -1\\ \end{bmatrix}