I am interested in solving the following differential equation: $$\frac{d\mathbf{v}}{dt}=A\mathbf{w}, \qquad \mathbf{v}=\left[x,y\right]^T, \mathbf{w}=[x^2,y^2,xy]^T, A\in\mathbb{R}^{2\times 3}$$
Equivalently, for $a_{ij}\in\mathbb{R}, \quad\forall i,j$ $$\frac{dx}{dt} = a_{11}x^2+a_{12}y^2+a_{13}xy$$ $$\frac{dy}{dt} = a_{21}x^2+a_{22}y^2+a_{23}xy$$
I have noticed that $$\mathbf{vv^T}=\left[\begin{array}{cc}x^2&xy\\xy&y^2\end{array}\right]$$
I'm looking for analytical solution to this problem.