Differential equation — Separable DE positive solutions

Question

Solve for , expressing the answer as or with .

This practice question was inspired by Question 11(d) 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.Separate: .
Finish the calculation, then check that the result meets the question’s conditions.Integrate: .
State the final answer clearly in the form the question requested.Answer: , or equivalently with .

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

Verification: Differentiate to recover .

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