{\color{blue}y_{\infty,1}} =
Determine the stationary solutions of the ODE
y'= Cy^3 - (A+B)*C y^2 + A*B*Cy .
{\color{blue}y_{\infty,1}} =
{\color{red}y_{\infty,2}} =
{\color{black}y_{\infty,3}} =
For a stationary solution {\color{orange}y_{\infty}}, we must have
{\color{orange}y'_{\infty}} = 0.
We therefore need to find the zeros of the right-hand side of the ODE
Cy^3 - (A+B)*C y^2 + A*B*Cy.
By factoring, we obtain Cy_{\infty} (y_{\infty}^2 - (A+B) y_{\infty} +
A*B) = 0. This gives us {\color{orange}y_{\infty}} = 0 as a first solution.
The two additional nonzero solutions are the roots of the term in the brackets.
The solutions of the quadratic equation y_{\infty}^2 - (A+B) y_{\infty} +
A*B = 0 can be found directly using Vieta's formulas:
For y_{\infty}^2 +-A-B y_{\infty}
+A*B = (y_{\infty} - {\color{blue}y_{\infty,1}} ) (y_{\infty} -{\color{red}y_{\infty,2}})
one gets {\color{blue}y_{\infty,1}} + {\color{red}y_{\infty,2}} = {\color{orange}A+B} und
{\color{blue}y_{\infty,1}} \cdot {\color{red}y_{\infty,2}} = {\color{teal}A*B}.
We thus get
{\color{blue}y_{\infty,1}} = A und {\color{red}y_{\infty,2}}= B.