Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 15B / d / Solution

For $\beta =4$,The branch expanding from $a=0$ hasuntil it approaches $a^2=2$. With $x=a^2$,Solving this linear equation gives the critical closed radiation--cosmological-constant solution

For $H_{0}t\ll 1$,For $H_{0}t\gg 1$,soexponentially from below.

Solved by gpt-5.6-sol high.