Sketching phase portrait of an ellipse I have a system of linear ODE's as follows:
$$\frac{dx}{dt} = y, \frac{dy}{dt} = -4x$$
which has solution 
$$\begin{bmatrix}x\\y\end{bmatrix} = \alpha\begin{bmatrix}\cos2t\\-2\sin2t\end{bmatrix} + \beta\begin{bmatrix}\sin2t\\2\cos2t\end{bmatrix}$$
And I'm having some issues trying to sketch the phase portrait.
I know the eigenvalues are $-2i$ and $+2i$
Clearly this means the critical point $(0,0)$ is a "centre" of ellipses.
But I don't know how to find the indicative equation of one of the solution curves.
In other words, how do I find the shape of the ellipses? How do I know which of x,y is the major and minor axis of the ellipses?
Thanks.

$$
     \frac{d}{dt}\{ 4x^{2}+y^{2}\} = 8x\frac{d x}{dt}+2y\frac{dy}{dt}=8xy-8yx=0.
$$
Therefore, $4x^{2}+y^{2}=C$ is constant in time.
