I was trying to solve $$ \dot{x}=ax^2+bxy\tag{1}\label{eq1}$$ $$ \dot{y}=cy^2+dxy \tag{2}\label{eq2}$$ According to riccati equation's initial showing $$ \frac{d}{dt} \left( \begin{bmatrix} M(t) \\ N(t) \end{bmatrix} \right) = \begin{bmatrix} -A&-B\\ C&D \end{bmatrix} \begin{bmatrix} M(t) \\ N(t) \end{bmatrix} where \ P=N(t)M(t)^{-1}, then \ \dot{P}= PBP + AP + PD + C \tag{3}\label{eq3}$$ To match \eqref{eq1} and \eqref{eq2} with the form \eqref{eq3}, I applied small change as follows, $$ \frac{d}{dt} \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} x& 0\\ 0& y \end{bmatrix} \begin{bmatrix} a& b\\ c& d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} \tag{4}\label{eq4}$$ and changed to the following $$ \frac{d}{dt} \left( \begin{bmatrix} x& 0\\ 0& y \end{bmatrix} \right)\begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} x& 0\\ 0& y \end{bmatrix} \begin{bmatrix} a& b\\ c& d \end{bmatrix} \begin{bmatrix} x& 0\\ 0& y \end{bmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix}\tag{5}\label{eq5} $$ This form looks similar to \eqref{eq3} $$ \dot{P}=PBP \tag{6}\label{eq6} $$ Therefore, I transformed \eqref{eq6} to \eqref{eq3} $$ \frac{d}{dt} \left( \begin{bmatrix} M(t) \\ N(t) \end{bmatrix} \right) = \begin{bmatrix} 0&-B\\ 0&0 \end{bmatrix} \begin{bmatrix} M(t) \\ N(t) \end{bmatrix} where \ X=N(t)M(t)^{-1} \tag{7}\label{eq7} \ and \ \, H=\begin{bmatrix} 0&-B\\ 0&0 \end{bmatrix}$$
Unlike riccati matrix equation, H in \eqref{eq7} is singular. So, directly applying riccati method seems not possible. Therefore, I chose exponential matrix of H. $$ \begin{bmatrix} M(t) \\ N(t) \end{bmatrix} = e^{Ht} \begin{bmatrix} M(0) \\ N(0) \end{bmatrix} , where \ X=N(t)M(t)^{-1} \tag{8}\label{eq8} $$ and I multiply $$ \begin{bmatrix} 1 \\ 1 \end{bmatrix}$$ to X in \eqref{eq8}.$$
Ok, I tested the result at the 1 dimension such as $$\dot{x} =ax^2$$ I showed well.
However, when I applied it into 2 dimensional problem that I wondered such as \eqref{eq1} \eqref{eq2}, It does not satisfied.
Could you let me know why and which part of my assumption was incorrect?
Such as $$\begin{bmatrix} x& 0\\ 0& y \end{bmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix}$$ or etc.