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2021 HSC Mathematics Extension 1

Question 12(a):
Differential equation

1 markdifferential equations

The direction field for a differential equation is given (on the writing booklet).

The graph of a particular solution to the differential equation passes through the point .

On the diagram provided, sketch the graph of this particular solution.

The supplied direction field with the initial point P marked.
The supplied direction field with the initial point P marked.

How to recognise this question

Question type: differential equations

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

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. A solution curve must pass through the marked point and be everywhere tangent to the local direction-field elements.
  2. Draw a smooth curve through that follows the slope marks; do not cross the short tangent segments transversely.
  3. NESA: 1 mark for a correct sketch. An acceptable solution curve does not cross any tangent line in the direction field.

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

On the direction field for , sketch the solution through . Describe the curve.

Direction field for $\dfrac{dy}{dx}=x$ with the initial point $P(0,1)$.
Direction field for with the initial point .
See the step-by-step answer guide →

Question 2

For , describe the solution through on a direction field.

Direction field for $\dfrac{dy}{dx}=y$ with the initial point $P(0,2)$.
Direction field for with the initial point .
See the step-by-step answer guide →

Question 3

State two properties that any correctly drawn particular solution on a direction field must satisfy.

See the step-by-step answer guide →