I'm interested in a solution to the initial value problem of the following quadratic dynamical system with 2 states $\bigl(x(t), y(t)\bigl)$:
$$ \begin{pmatrix} \frac{dx}{dt} \\ \frac{dy}{dt}\\ \end{pmatrix} = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \begin{pmatrix} x^2 \\ y\\ \end{pmatrix} $$ It would be a linear dynamical system if it were not for the quadratic term $x^2$ in the right hand side.
The form of the differential equation is analogous to the Riccati equation $y'=a+by+cy^{2}$ but generalized to a system of differential equations. I have seen that there is a matrix version of the Riccati equation but I haven't been able to put my problem into that form.
Is there an analytical solution (or approximation) to the initial value problem for such a quadratic dynamical system?