Differential equation — Choose the correct branch

Question

Solve given .

This practice question was inspired by Question 13(a) in the 2025 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..
Simplify the previous line carefully, keeping exact values where possible.; using gives .
Simplify the previous line carefully, keeping exact values where possible..
Finish the calculation, then check that the result meets the question’s conditions.The initial value is negative, so choose .
State the final answer clearly in the form the question requested.Final answer:

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

Verification: At the solution is , and differentiating gives .

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