{\color{red}y_{\infty,1}} =
{\color{blue}y_{\infty,2}} =
The vector field
F: \mathbb R^2 \to \mathbb R^2,
\begin{pmatrix} y_1 \\ y_2 \end{pmatrix}\mapsto
\begin{pmatrix} Ay_1 + By_2 \\
Cy_1^2 + Dy_1 y_2^2 \end{pmatrix}
defines a system y' = F(y).
Find the fixed point
y_{\infty} = \left({\color{red}y_{\infty,1}},{\color{blue}y_{\infty,2}}\right) \neq (0,0).
{\color{red}y_{\infty,1}} =
{\color{blue}y_{\infty,2}} =
Looking for y_{\infty} = \left({\color{red}y_{\infty,1}},{\color{blue}y_{\infty,2}}\right) \neq (0,0), i.e.
solutions of F(y) =0.
The first equation is Ay_1 + By_2 = 0 and therefore
{\color{red}y_1 = fractionReduce(-B,A) y_2}.
The second equation is
0= C{\color{red}y_1^2} + D{\color{red}y_1} y_2^2 =
fractionReduce(C*B*B,A*A)y_2^2 - fractionReduce(B*D,A) y_2^3
and thus because y_1 \neq 0 \neq y_2 it follows
{\color{blue}y_2 = fractionReduce(C*B,D*A)}.
Besides the origin there is another fixed point
\displaystyle
y_{\infty} = \left({\color{red}y_{\infty,1}},{\color{blue}y_{\infty,2}}\right)
= \left(fractionReduce(-(C*B*B),D*A*A), fractionReduce(C*B,D*A)\right).