Question 7
The differential equation for which the solution is a logistic equation of the form $y=\dfrac{a}{b+Ce^{-at}}$, where $a$, $b$ and $C$ are constants, is
Not quite. The right-hand side depends only on $t$, so integrating gives a quadratic in $t$. That isn’t a logistic curve. In 2023, 33.5% of Queensland students chose (A).
Not quite. This has the $(1-0.01y)$ factor, but no factor of $y$. Its solution creeps towards $100$ with no S-shaped rise, so it isn’t logistic. In 2023, 17.7% of Queensland students chose (B).
Not quite. The right-hand side depends only on $t$, so the solution is a cubic in $t$. In 2023, 32.3% of Queensland students chose (C).
That’s it. Step through the hints below to see why. In 2023, 15.5% of Queensland students chose (D).
Each curve starts at $y=5$ when $t=0$. Dashed: $y=100$.
An S-shaped curve: slow growth, then fast, then levelling off at $100$. That is the logistic shape.
The rate is biggest at the start and falls as $y$ grows, so the curve bends the opposite way to an S. It does approach $100$, but very slowly. Over this window it only reaches about $14$.
The right-hand side depends only on $t$, so the solution is a polynomial that doesn’t level off at a limit.
The logistic equation $y=\dfrac{a}{b+Ce^{-at}}$ is the solution to a differential equation of this form.
$$\frac{dy}{dt}=y(a-by)$$
So the right-hand side has to depend on $y$, not on $t$. That rules out (A) and (C).
It also needs the factor $y$ in front. Only (D) has both. Expand the bracket to match $y(a-by)$: $0.25y(1-0.01y)=y(0.25-0.0025y)$, so $a=0.25$ and $b=0.0025$.
The logistic equation $y=\frac{a}{b+Ce^{-at}}$ is the solution to $\frac{dy}{dt}=y(a-by)$.
$$\frac{dy}{dt}=0.25y(1-0.01y)=y(0.25-0.0025y)\quad\text{so }a=0.25,\ b=0.0025$$
How Queensland students answered in 2023
More students chose (A) than the answer. From the QCAA Specialist Mathematics subject report for 2023. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2023 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2023, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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