Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 2 / 2C (Differential Equations)

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

The fixed points are $x=-2\pm 2\sqrt{1+b}$, so they exist for $b\ge -1$. A fixed point of an iteration is stable when $|g^{\prime }(x)|<1$; here this is $|x|<2$. Only the plus root can satisfy it, and it does precisely for