2022 HSC Mathematics Extension 1
Question 14(a):
Differential equation
Find the particular solution of the differential equation that passes through the point .
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
- Separate (for , ): .
- Rewrite .
- Integrate: .
- So with , i.e. .
- Through : .
- Particular solution: , valid on an interval containing and excluding (e.g. ).
- NESA: 4 marks; 3 for correct equation with constant; 2 for correct primitive; 1 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 .
Question 2
Find the general solution of .
Question 3
Show that satisfies and .