2024 HSC Mathematics Extension 1
Question 14(a):
Differential equation
Find the domain and range of the function that is the solution to the differential equation and whose graph passes through the origin.
How to recognise this question
Question type: differential equations
- The command and notation point to differential equations.
- The requested response is a worked solution.
How to handle it: separate or integrate the differential equation, include the constant, then use the initial condition.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- Rewrite and separate: .
- Integrate: .
- Through origin : .
- So .
- Domain: require , and the solution is defined for all such (as , ). Domain .
- Range: as , from above; as , . Also so is increasing. Range .
- NESA 14(a): 4 marks correct; 3 marks for correct domain or range; 2 marks for integrated relation with constant; 1 mark for separating variables.
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 4 total marks for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
Solve through , then find the domain and range of the solution function.
Question 2
Find the explicit solution of with , and state its maximal domain.
Question 3
Given on , determine the range.