Hi all,
I have been doing some ODE's and came across this question:
Given, \frac{dy}{dt} = k + 2y - y^2
Locate the equilibrium solutions and determine their type for all values of k, including
any bifurcation values.

so first I find the equilibrium solutions:
\frac{-2\pm\sqrt{4-4(-1)(k)}}{-2}
and find the solutions:
y = -\sqrt{k+1}+1, \sqrt{k+1}+1

I find the partial derivative with respect to t:
\frac{\partial u}{\partial t} = -2y + 2
and check solution type:
\left. \frac{\partial y}{\partial t} \right|_{t=-\sqrt{k+1}+1} = 2\sqrt{k+1}, y = -\sqrt{k+1}+1 is a sink if 2\sqrt{k+1} < 0 (No solution) and a source when 2\sqrt{k+1} > 0, (x > -1)
similarly,
\left. \frac{\partial y}{\partial t} \right|_{t=\sqrt{k+1}+1} = -2\sqrt{k+1}, y = \sqrt{k+1}+1 is a sink if -2\sqrt{k+1} < 0, (x > -1) and a source when -2\sqrt{k+1} > 0 (No solution)

So, it is clear that y = -\sqrt{k+1}+1 is a source, and y = \sqrt{k+1}+1 is a sink.

I am not sure if my working is right?, If so how do I go ahead and find the bifurcation value? Thanks