Find steady states of the differential equation

Question : Find steady states of the differential equation dy/dx =y(1-y)-a=f(y) where a is any parameter.

Solution :

\[\frac{dy}{dx}=y\left(1-y\right)-a=f(y) \]

(1)

Where a is any parameter.

We shall obtain Steady states of the given differential equation as follows

We shall solve \[\ f\left(y\right)=0\] \[y\left(1-y\right)-a=0\] \[y^2-y+a=\ 0\] \[y=\frac{1\pm\sqrt{1-4a}}{2}\] \[y_1=\frac{1+\sqrt{1-4a}}{2}\ \ ,\ \ \ y_2=\ \frac{1-\sqrt{1-4a}}{2}\] The values of \[y_1,\ y_2\] will exist if and only if \[\ 1-4a>\ 0\ \] or\[ a<\frac{1}{4} \] And steady states will be given as \[y_1=\frac{1+\sqrt{1-4a}}{2}\ \ ,\ \ \ y_2=\ \frac{1-\sqrt{1-4a}}{2}\]