system nonlinear differential equation

the problem says:
"show that the zero solution is nonlinear stable. For this, find the change of variable that transforms this system in a linear system"

[math] \frac{dx}{dt}=-x + \beta (x^2+ y^2) [/math]

[math] \frac{dy}{dt}=-2y + \gamma x y [/math]

i tried with the method of eigenvalues of Jacobian matrix, and both eigenvalues are negatives, but my teacher says that this method is incorrect. Help please