Differential equation — Alternate equilibrium multiparts

Question

For :

(i) Explain why a solution with can never reach .

(ii) Describe the long-term behaviour of the solution through .

(iii) Find the population at which the growth rate is largest, using calculus.

This practice question was inspired by Question 13(a) in the 2024 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: separate or integrate the differential equation, include the constant, then use the initial condition.(i) is an equilibrium; autonomous uniqueness prevents solutions from crossing it. Since , the solution through cannot reach .
Simplify the previous line carefully, keeping exact values where possible.(ii) The solution increases and approaches asymptotically from below.
Finish the calculation, then check that the result meets the question’s conditions.(iii) Let . Then . , so maximum.
State the final answer clearly in the form the question requested.Answer: (i) cannot cross equilibrium ; (ii) ; (iii) .

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: Half carrying capacity maximises logistic growth: .

Any mark labels are a TestMum study aid, not an official NESA marking allocation.