IB Maths AI HL AI 5.3 Differential equations questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

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Question 1

Paper 2Medium7 marks

A laboratory monitors a culture of cells. Its population, $y$ is measured in thousands of cells, and $t$ is measured in hours. The rate of change of the population is modelled by

\frac{\mathrm{d}y}{\mathrm{d}t}=0.06y(14-y).

The table summarizes the slopes in the slope field for this model. Each entry is the value of $\frac{\mathrm{d}y}{\mathrm{d}t}$ at the point given by its row and column. Slopes are in thousands of cells per hour.

$y$	$t=0$	$t=0.5$	$t=1$
0	0	0	0
7	2.94	2.94	2.94
11	1.98	1.98	1.98
14	0	0	0
18	$-4.32$	$-4.32$	$-4.32$

(a)

Show that the slope of the field at $(t,y)=(0,9)$ is $2.7$

[2]

(b)

A culture has $y=9$ at $t=0$ Use the gradient at this point to estimate its population at $t=0.5$

[2]

(c)

Use the slope field to decide whether your estimate in part (b) is likely to be greater or less than the modelled population at $t=0.5$ Justify your answer, and state the population towards which the field predicts the culture will tend over a long time.

[3]

Question 2

Paper 2Medium7 marks

In a reserve, the prey and predator populations are represented by indices $x$ and $y$ measured in hundreds of animals. The time $t$ is measured in years after monitoring begins. The populations are modelled by

\frac{\mathrm{d}x}{\mathrm{d}t}=0.8x-0.04xy, \qquad \frac{\mathrm{d}y}{\mathrm{d}t}=0.02xy-0.6y,

with $x(0)=20$ and $y(0)=5$ A researcher uses Euler’s method with a step size of $0.5$ years.

(a)

Show that the Euler estimate for $x(0.5)$ is $26$ and find the corresponding estimate for $y(0.5)$

[3]

(b)

Hence, use one further Euler step to estimate $x(1)$ and $y(1)$

[2]

(c)

A researcher claims that the predator population is decreasing at $t=1$ since the estimate for $y(1)$ is lower than that for $y(0.5)$ Use the model to assess this claim.

[2]

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