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2022 HSC Mathematics Extension 1

Question 10:
Differential equation

1 markdifferential equations

Which of the following could be the graph of a solution to the differential equation ?

All four candidate solution graphs supplied with the original differential-equation question.
All four candidate solution graphs supplied with the original differential-equation question.
  1. A family of branches with vertical asymptotes, including decreasing segments
  2. A single smooth increasing curve lying between two horizontal asymptotes (never decreasing)
  3. A curve with a clear local maximum (so it decreases on an interval)
  4. A curve that is increasing overall but flattens and then steepens repeatedly without horizontal bounds matching equilibria

How to recognise this question

Question type: differential equations

  • The command and notation point to differential equations.
  • The requested response is a multiple choice.

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

  1. Note that for all real , with equality precisely when , i.e. .
  2. Therefore every solution is non-decreasing: the graph can never have a negative slope.
  3. Option A includes decreasing pieces; option C has a local maximum — both impossible.
  4. Option D may look non-decreasing but does not respect the equilibrium levels where slope must be exactly zero when hits ; typical incorrect sketches cross or ignore those lines.
  5. Option B is non-decreasing and approaches horizontal asymptotes consistent with the zero-slope levels of the autonomous equation.
  6. Answer: B. (NESA multiple-choice key: B.)

Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.

The paper records 1 total mark 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

For , which property must every solution graph have?

  1. Never strictly decreasing on an interval where
  2. Always oscillatory
  3. Always has a vertical asymptote
  4. Must decrease somewhere
See the step-by-step answer guide →

Question 2

Solutions of can have horizontal asymptotes at

  2. only
  3. only
  4. no horizontal lines
See the step-by-step answer guide →

Question 3

A proposed solution graph of falls from to as increases. This is

  1. impossible, because on
  2. possible near an equilibrium
  3. required by uniqueness
  4. possible if is negative
See the step-by-step answer guide →