Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 2 / 2C / a / Solution

The fixed points are $0$ and, when $a\ge -1$, $1\pm \sqrt{1+a}$. For a scalar autonomous equation, a simple fixed point is stable when $f^{\prime }<0$. Since $f^{\prime }(0)=a$ and $f^{\prime }(r)=2r(1-r)$ at either nonzero root, excluding the bifurcation values as requested,and

Solved by gpt-5.6-sol high.