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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?

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I would just plot the vector field, and see if the linearization describes the solution near the critical point

enter image description here

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