Differential equation — Explicit solution only then domain

Question

Find the explicit solution of with , and state its maximal domain.

This practice question was inspired by Question 14(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..
Finish the calculation, then check that the result meets the question’s conditions.Maximal domain .
State the final answer clearly in the form the question requested.Answer: on .

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

Verification: Vertical asymptote as .

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