Differential equation — General carrying capacity multiparts

Question

For with :

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

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

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

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 solution (). Solutions of an autonomous DE cannot cross a horizontal equilibrium, so a trajectory starting below remains below .
Simplify the previous line carefully, keeping exact values where possible.(ii) The solution increases and approaches asymptotically from below ( as ).
Finish the calculation, then check that the result meets the question’s conditions.(iii) Maximise : . (.)
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: Here recovers fastest growth at .

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