# Does the linearized system accurately describe the local behavior near the equilibrium points

For this nonlinear system, does the linearized system accurately describe the local behavior near the equilibrium points?

\begin{cases} x' = x + y^2 \\\\ y' = 2y \\\\ \end{cases}

The nonlinear system has an equilibrium point at $(0, 0)$ and so I linearize this nonlinear system into \begin{cases} x' = x \\\\ y' = 2y \\\\ \end{cases}

and essentially solving $AX = X'$ where $\displaystyle A = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix},$ which I find the general solution to be $\displaystyle \alpha e^{2t}[0, 1]^T + \beta e^t[1, 0]^T$ where $\alpha$ and $\beta$ are arbitrary constants. But how do I know if this linearized system accurately describes the local behavior near the equilibrium points or not?