Differential Equations - IB Math AI Question Bank

Question 1

HL • Paper 1

Easy

Calculator Permitted

The slope field for the differential equation

\frac{\mathrm{d}y}{\mathrm{d}x}=y(2-y)

is shown for $-2\leq x\leq 3$ and $-1\leq y\leq 3$ .

A slope field on a rectangular coordinate grid for the stated differential equation. It must show horizontal bands of equal slope, including horizontal segments along the two equilibrium levels, and sufficient space to sketch a solution through the point $(0,1)$. No solution curve is included.

A

State the two equilibrium solutions.

[2]

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B

State whether the gradient is positive or negative in the region $0<y<2$ .

[1]

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C

On the slope field, sketch the solution curve passing through $(0,1)$ .

[2]

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0

Question 2

HL • Paper 1

Easy

Calculator Permitted

The function $y$ satisfies

\frac{\mathrm{d}y}{\mathrm{d}x}=x-y,\qquad y(0)=1

Euler's method with step size $h=0.2$ is used to estimate $y(0.6)$ .

A

Write down the Euler recurrence for $y_{n+1}$ in terms of $x_n$ and $y_n$ .

[1]

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B

Find the Euler estimates for $y(0.2)$ and $y(0.4)$ .

[2]

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C

Hence estimate $y(0.6)$ .

[1]

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D

The exact value is $y(0.6)=0.698$ , correct to three significant figures. State whether the Euler estimate is an overestimate or an underestimate.

[1]

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0

Question 3

HL • Paper 1

Medium

Calculator Permitted

A medicine is eliminated from a patient's bloodstream at a rate proportional to the mass $Q$ of medicine present. Initially, the bloodstream contains $80$ mg of the medicine. After $3$ hours, $50$ mg remains.

A

Write down a differential equation that models the mass of medicine, where $t$ is measured in hours and $k$ is a positive constant.

[1]

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B

Find $k$ .

[3]

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C

Find the mass of medicine remaining after $8$ hours.

[2]

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0

Question 4

HL • Paper 1

Medium

Calculator Permitted

The area $G$ covered by algae, in square metres, is modelled by

\frac{\mathrm{d}G}{\mathrm{d}t}=k\sqrt{G}

where $t$ is the time in days and $k>0$ . Initially, $G=4$ , and after $6$ days, $G=25$ .

A

By solving the differential equation, show that $2\sqrt{G}=kt+4$ .

[3]

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B

Find $k$ .

[1]

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C

Determine the time at which the area covered by algae first reaches $49\ \text{m}^2$ .

[2]

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0

Question 5

HL • Paper 1

Medium

Calculator Permitted

A spherical object expands so that its radius $r$ , in centimetres, increases at a rate inversely proportional to its radius. Initially, its radius is $4$ cm, and after $6$ seconds its radius is $8$ cm.

A

Write down a differential equation for $r$ in terms of $t$ , using a positive constant $k$ .

[1]

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B

Solve the differential equation and hence find $k$ .

[3]

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C

Find the time at which the radius reaches $12$ cm.

[2]

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0

Question 6

HL • Paper 1

Medium

Calculator Permitted

The function $y$ satisfies

\frac{\mathrm{d}y}{\mathrm{d}x}=y(2-y),\qquad y(0)=0.5

Two Euler approximations for $y(0.5)$ are calculated.

A

Using one step of length $0.5$ , find an estimate for $y(0.5)$ .

[2]

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B

Using two steps of length $0.25$ , find another estimate for $y(0.5)$ .

[2]

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C

The exact value of $y(0.5)$ is $0.951$ , correct to three significant figures. State which Euler approximation is more accurate.

[1]

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0

Question 7

HL • Paper 1

Medium

Calculator Permitted

The populations $x$ and $y$ , in hundreds, of two interacting species are modelled by

\begin{aligned} \frac{\mathrm{d}x}{\mathrm{d}t}&=0.4x-0.02xy,\\ \frac{\mathrm{d}y}{\mathrm{d}t}&=-0.3y+0.01xy. \end{aligned}

Initially, $x=20$ and $y=10$ . Euler's method is used with step size $h=0.5$ .

A

Write down the Euler recurrence relations for $x_{n+1}$ and $y_{n+1}$ .

[2]

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B

Find the estimates for $x$ and $y$ when $t=0.5$ .

[2]

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C

Hence estimate $x$ and $y$ when $t=1$ .

[2]

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0

Question 8

HL • Paper 2

Medium

Calculator Permitted

An infection is modelled by the coupled system

\begin{aligned} \frac{\mathrm{d}S}{\mathrm{d}t}&=-0.002SI,\\ \frac{\mathrm{d}I}{\mathrm{d}t}&=0.002SI-0.4I,\\ \frac{\mathrm{d}R}{\mathrm{d}t}&=0.4I, \end{aligned}

where $S$ , $I$ and $R$ are population sizes. Initially, $(S,I,R)=(90,10,0)$ . Euler's method is used with $h=0.1$ .

A

Calculate the Euler estimates for $S$ , $I$ and $R$ when $t=0.1$ .

[3]

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B

Hence calculate the Euler estimates for $S$ , $I$ and $R$ when $t=0.2$ .

[2]

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0

Question 9

HL • Paper 1

Medium

Calculator Permitted

A phase portrait is associated with the system

\begin{aligned} \frac{\mathrm{d}x}{\mathrm{d}t}&=3x+y,\\ \frac{\mathrm{d}y}{\mathrm{d}t}&=x+3y. \end{aligned}

A

Find the velocity vector at the point $(1,0)$ .

[1]

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B

Find the eigenvalues and corresponding eigendirections of the system.

[3]

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C

Describe the long-term behaviour of a non-equilibrium trajectory.

[1]

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0

Question 10

HL • Paper 1

Medium

Calculator Permitted

Consider the system

\frac{\mathrm{d}}{\mathrm{d}t} \begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}-1&-2\\2&-1\end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix}

A

Find the eigenvalues of the coefficient matrix.

[2]

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B

Describe the form and stability of trajectories near the origin.

[1]

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C

State whether the trajectories rotate clockwise or anticlockwise.

[1]

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0

Question 11

HL • Paper 1

Medium

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The displacement $x$ of a particle satisfies

\frac{\mathrm{d}^2x}{\mathrm{d}t^2}+2\frac{\mathrm{d}x}{\mathrm{d}t}+5x=0

Initially, $x=1$ and $\dfrac{\mathrm{d}x}{\mathrm{d}t}=0$ .

A

By defining $v=\dfrac{\mathrm{d}x}{\mathrm{d}t}$ , write the equation as a coupled system of first-order differential equations.

[2]

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B

Write down the Euler recurrence relations for $x$ and $v$ using step size $h=0.1$ .

[2]

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C

Use Euler's method to estimate the displacement and velocity when $t=0.2$ .

[2]

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0

Question 12

HL • Paper 1

Medium

Calculator Permitted

Solutions of the differential equation

\frac{\mathrm{d}y}{\mathrm{d}x}=x^2-y

may have stationary points.

A

Find the equation of the curve on which all stationary points lie.

[1]

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B

Determine whether a stationary point with $x<0$ is a local maximum or a local minimum.

[2]

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C

Classify the stationary point $(2,4)$ .

[1]

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0

Question 13

HL • Paper 1

Medium

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Consider the linear system

\frac{\mathrm{d}}{\mathrm{d}t} \begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}4&1\\-2&1\end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix}

A

Find the eigenvalues of the coefficient matrix.

[2]

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B

Find an eigenvector corresponding to each eigenvalue.

[2]

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C

Classify the equilibrium at the origin and state its stability.

[1]

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0

Question 14

HL • Paper 1

Medium

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The displacement $x$ of an object satisfies

\frac{\mathrm{d}^2x}{\mathrm{d}t^2}-\frac{\mathrm{d}x}{\mathrm{d}t}-6x=0

with $x(0)=2$ and $x'(0)=1$ .

A

Find the roots of the characteristic equation.

[2]

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B

Hence find the particular solution for $x(t)$ .

[3]

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C

Find the displacement when $t=0.5$ .

[2]

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0

Question 15

HL • Paper 1

Medium

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The displacement $x$ of a damped system satisfies

\frac{\mathrm{d}^2x}{\mathrm{d}t^2}+4\frac{\mathrm{d}x}{\mathrm{d}t}+13x=0

Let $v=\dfrac{\mathrm{d}x}{\mathrm{d}t}$ .

A

Write the corresponding matrix system for $\begin{pmatrix}x\\v\end{pmatrix}$ .

[2]

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B

Find the eigenvalues of the coefficient matrix.

[1]

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C

Describe the phase portrait near the origin, including its direction of rotation.

[2]

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0

Question 16

HL • Paper 1

Medium

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The vector $\mathbf{X}(t)=\begin{pmatrix}x(t)\\y(t)\end{pmatrix}$ satisfies

\frac{\mathrm{d}\mathbf{X}}{\mathrm{d}t}= \begin{pmatrix}2&1\\1&2\end{pmatrix}\mathbf{X}

with $\mathbf{X}(0)=\begin{pmatrix}3\\1\end{pmatrix}$ .

A

Find the two eigenvalues of the coefficient matrix.

[2]

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B

Find an eigenvector corresponding to each eigenvalue.

[2]

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C

Hence find $x(t)$ and $y(t)$ .

[3]

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0

Question 17

HL • Paper 2

Medium

Calculator Permitted

Water leaks from a tank. The volume $V$ , in litres, decreases at a rate proportional to $\sqrt{V}$ . Initially the tank contains $900$ litres, and after $4$ hours it contains $676$ litres. Let $t$ be the time in hours and let $k>0$ .

A

I.

Write down a differential equation modelling the volume of water.

[1]

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II.

By separating variables, show that $\sqrt V=30-\dfrac{k}{2}t$ .

[3]

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B

I.

Find $k$ .

[2]

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II.

Hence determine when the volume first reaches $400$ litres.

[2]

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0

Question 18

HL • Paper 2

Medium

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A function $y$ satisfies

\frac{\mathrm dy}{\mathrm dx}=xe^{-y},\qquad y(0)=0

A

I.

Find the particular solution in the form $y=f(x)$ .

[4]

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II.

Write down the exact value of $y(2)$ .

[1]

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B

Find the positive values of $x$ for which the gradient of the solution curve is $0.5$ .

[3]

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0

Question 19

HL • Paper 2

Medium

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The function $y$ satisfies

\frac{\mathrm dy}{\mathrm dx}=\sin x-0.4y^2, \qquad y(0)=0.5

A

I.

Write down the Euler recurrence with step size $h=0.25$ .

[2]

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II.

Use this recurrence to estimate $y(1)$ .

[3]

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B

Using Euler's method with $h=0.5$ gives $y(1)\approx0.649$ . Calculate the percentage difference between the two Euler estimates, relative to the estimate using $h=0.25$ .

[3]

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0

Question 20

HL • Paper 3

Medium

Calculator Permitted

The salt concentration $C$ , in grams per litre, in a lagoon is modelled by assuming that its rate of increase is proportional to the difference between $C$ and the limiting concentration $12$ . Initially, $C=2$ , and after three days $C=7$ . Time $t$ is measured in days.

A

Set up the differential-equation model.

I.

Write down the differential equation, using a positive constant $k$ .

[1]

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II.

State the units of $k$ .

[1]

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B

By separation of variables, show that $C=12-10e^{-kt}$ .

[3]

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C

Use the model to determine the first time at which $C=10$ .

[3]

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0

Question 21

HL • Paper 2

Hard

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The population $P$ of an insect colony is modelled by

\frac{\mathrm dP}{\mathrm dt}=0.3P\left(1-\frac{P}{1200}\right)

where $t$ is measured in days. Initially, $P=200$ .

A

I.

Show that the differential equation may be separated in the form

\left(\frac1P+\frac1{1200-P}\right)\,\mathrm dP=0.3\,\mathrm dt

[2]

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II.

Hence find the particular solution for $P$ in terms of $t$ .

[4]

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B

Determine the time at which the population reaches $900$ , and hence state when it exceeds $900$ .

[3]

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0

Question 22

HL • Paper 2

Hard

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The slope field shown represents

\frac{\mathrm dy}{\mathrm dx}=(x-1)(y+1)

Slope field for dy/dx=(x-1)(y+1), with the point (0,0) marked.

A

I.

State the equilibrium solution.

[1]

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II.

For a non-equilibrium solution, explain why a stationary point occurs at $x=1$ and classify it when $y>-1$ .

[3]

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B

The solution curve passes through $(0,0)$ . Find its minimum value.

[4]

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0

Question 23

HL • Paper 2

Hard

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Consider the differential equation

\frac{\mathrm dy}{\mathrm dx}=(2-x)(y-1)

A solution curve passes through $(0,3)$ .

Plot of the solution curve with y=1 and x=2 marked, plus the initial point.

A

I.

State the equilibrium solution and the vertical line on which all non-equilibrium solutions have stationary points.

[2]

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II.

Use the displayed solution-curve graph to classify the stationary point of the specified solution.

[2]

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B

Determine the coordinates of this stationary point.

[5]

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0

Question 24

HL • Paper 2

Hard

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Two connected tanks contain amounts $x$ and $y$ litres of a dissolved substance. The amounts are modelled by

\frac{\mathrm dx}{\mathrm dt}=-0.3x+0.1y, \qquad \frac{\mathrm dy}{\mathrm dt}=0.3x-0.2y

Initially, $x=40$ and $y=10$ , where $t$ is measured in hours.

A schematic of two connected tanks labelled $x$ and $y$, with transfer arrows in both directions and an outlet from the second tank. No numerical flow rates are displayed.

A

I.

Write down the simultaneous Euler recurrences for step size $h=0.5$ .

[2]

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II.

Estimate $x$ and $y$ when $t=1$ .

[4]

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B

Use the model to explain why the total amount $x+y$ always decreases when $y>0$ .

[3]

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0

Question 25

HL • Paper 2

Hard

Calculator Permitted

Consider the system

\frac{\mathrm d}{\mathrm dt} \begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}-0.5&-2\\2&-0.5\end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix}

with initial state $(4,0)$ .

Phase-plane trajectory from the initial state.

A

I.

Find the eigenvalues. Use the displayed phase-plane trajectory to describe the stability and form of the phase portrait.

[3]

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II.

Determine the direction of rotation.

[1]

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B

Use Euler's method with $h=0.1$ to estimate the state when $t=0.2$ .

[4]

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0

Question 26

HL • Paper 2

Hard

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The displacement $x$ , in metres, of a damped mechanical system satisfies

\frac{\mathrm d^2x}{\mathrm dt^2}+5\frac{\mathrm dx}{\mathrm dt}+6x=0

where $x(0)=3$ and $x'(0)=-1$ .

A

I.

By defining $v=x'$ , write the equation as a first-order matrix system and find its eigenvalues.

[3]

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II.

Hence find the particular solution for $x(t)$ .

[3]

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B

Calculate $x(0.5)$ and explain the long-term behaviour of the system.

[3]

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0

Question 27

HL • Paper 2

Hard

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The vertical displacement $x$ , in metres, of a ball moving upwards through a resisting medium satisfies

\frac{\mathrm d^2x}{\mathrm dt^2}=-0.4\frac{\mathrm dx}{\mathrm dt}-9.8

Initially, $x=0$ and $x'=20\ \mathrm{m\,s}^{-1}$ .

A

I.

Define $v=x'$ and write down Euler recurrence relations with step size $h=0.2$ .

[2]

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II.

Use Euler's method to estimate the displacement and velocity at $t=0.6$ .

[4]

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B

Without resistance, the displacement after $0.6$ seconds would be modelled by $x=20t-4.9t^2$ . Compare this value with the Euler estimate and interpret the difference.

[3]

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0

Question 28

HL • Paper 3

Hard

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A block of ice has volume $V$ cubic centimetres. During melting, its rate of decrease in volume is proportional to $V^{2/3}$ . Initially, its volume is $1000\ \text{cm}^3$ . After six minutes, its volume is $512\ \text{cm}^3$ .

A

Develop a model for the volume.

I.

Write down a differential equation for $V$ , using a positive constant $k$ .

[1]

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II.

Solve the differential equation to show that $V=(10-\frac{k}{3}t)^3$ while $V>0$ .

[3]

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B

Find $k$ and hence determine the time predicted for the block to melt completely.

[3]

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C

State one reason why the predicted melting time may not be reliable.

[1]

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0

Question 29

HL • Paper 3

Hard

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A slope field is shown for the differential equation

\frac{dy}{dx}=(x-1)(y-2)

y	x=-1	x=0	x=2
1	2	1	-1
2	0	0	0
3	-2	-1	1

A

Interpret the zero-gradient features of the field.

I.

State the two lines on which the field has horizontal segments.

[2]

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II.

Explain why only one of these lines is an equilibrium solution.

[1]

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B

Find the particular solution passing through $(0,3)$ .

[3]

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C

Hence determine and classify the stationary point of this solution.

[2]

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0

Question 30

HL • Paper 3

Hard

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The number $P$ of users who have received an emergency alert is modelled by

\frac{dP}{dt}=0.6P\left(1-\frac{P}{5000}\right),\qquad P(0)=500

where $t$ is measured in hours.

A

Euler's method is used with step size $h=0.5$ hours.

I.

Write down the Euler recurrence.

[1]

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II.

Find the Euler estimate for $P(1)$ .

[2]

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B

By separation of variables, the exact solution can be written as $P=\dfrac{5000}{1+Ae^{-0.6t}}$ . Find $A$ .

[2]

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C

Compare the Euler estimate with the exact value at $t=1$ , and comment on the effect of using a smaller step size.

[3]

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0

Question 31

HL • Paper 3

Hard

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Two connected tanks contain amounts $x$ and $y$ kilograms of a dissolved mineral. The transfer process is modelled by

\frac{dx}{dt}=-0.3x+0.1y, \qquad \frac{dy}{dt}=0.3x-0.1y

where $t$ is measured in hours. Initially, $(x,y)=(20,0)$ .

A diagram of two connected tanks labelled $x$ and $y$, with arrows showing mineral solution transferring in both directions and no mineral entering or leaving the two-tank system.

A

Investigate conservation in the model.

I.

Find $\dfrac{d}{dt}(x+y)$ .

[1]

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II.

Interpret this result in context.

[1]

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B

Use Euler's method with $h=1$ to estimate $(x,y)$ after two hours.

[3]

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C

Determine the equilibrium amounts consistent with the initial total amount.

[2]

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D

Explain why the Euler values preserve the total amount exactly in this model.

[1]

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0

Question 32

HL • Paper 3

Hard

Calculator Permitted

Deviations $x$ and $y$ from the target values of two economic indicators are modelled by

\frac{d}{dt}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}-1&2\\-2&-1\end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix}

A

Analyse the coefficient matrix.

I.

Find its eigenvalues.

[2]

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II.

Classify the equilibrium at the origin.

[1]

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B

Determine the direction of rotation and justify your answer using the point $(1,0)$ .

[2]

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C

Sketch a representative phase portrait, including arrows and the long-term behaviour.

[3]

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0

Question 33

HL • Paper 3

Hard

Calculator Permitted

The displacement $x$ , in metres, of a damped platform satisfies

\frac{d^2x}{dt^2}+5\frac{dx}{dt}+6x=0

with $x(0)=0.1$ and $x'(0)=0$ .

A

Construct an Euler approximation using $v=x'$ .

I.

Write the coupled first-order system and Euler recurrences for step size $h=0.1$ .

[2]

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II.

Estimate $x(0.2)$ and $v(0.2)$ .

[2]

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B

Find the exact displacement $x(t)$ .

[3]

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C

Compare the Euler estimate of $x(0.2)$ with the exact value.

[1]

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0

Question 34

HL • Paper 2

Hard

Calculator Permitted

Temperatures $x$ and $y$ , in degrees Celsius, in two connected chambers are modelled by

\frac{\mathrm dx}{\mathrm dt}=0.2(100-x)-0.1(x-y), \qquad \frac{\mathrm dy}{\mathrm dt}=0.1(x-y)-0.05y

Initially, $(x,y)=(20,0)$ and $t$ is measured in minutes.

A

I.

Using Euler's method with $h=1$ , find the estimate after one minute.

[3]

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II.

Hence estimate the temperatures after two minutes.

[3]

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B

Determine the equilibrium temperatures predicted by the model.

[4]

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0

Question 35

HL • Paper 2

Hard

Calculator Permitted

The vector $\mathbf X(t)=\begin{pmatrix}x(t)\\y(t)\end{pmatrix}$ satisfies

\frac{\mathrm d\mathbf X}{\mathrm dt}= \begin{pmatrix}-3&1\\1&-3\end{pmatrix}\mathbf X, \qquad \mathbf X(0)=\begin{pmatrix}6\\2\end{pmatrix}

A

I.

Find the eigenvalues and a corresponding eigenvector for each eigenvalue.

[3]

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II.

Hence find the particular solution.

[3]

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B

Determine the time when $y=0.9x$ , and describe the trajectory as $t\to\infty$ .

[4]

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0

Question 36

HL • Paper 2

Hard

Calculator Permitted

A coupled system is given by

\frac{\mathrm d}{\mathrm dt} \begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}1&2\\2&1\end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix}, \qquad (x(0),y(0))=(3,-1)

A

I.

Find the eigenvalues and corresponding eigendirections, and classify the origin.

[3]

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II.

Find the particular solution.

[2]

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B

Find when the trajectory first crosses the positive $x$ -axis, and state its asymptote as $t\to\infty$ .

[4]

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0

Question 37

HL • Paper 2

Hard

Calculator Permitted

The system

\frac{\mathrm dx}{\mathrm dt}=-3y, \qquad \frac{\mathrm dy}{\mathrm dt}=12x

has initial condition $(x(0),y(0))=(2,0)$ .

A

I.

Find the eigenvalues and classify the origin.

[2]

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II.

Show that $4x^2+y^2$ is constant along every trajectory.

[3]

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B

Hence find the maximum value of $|y|$ , the direction of rotation and the period of the motion.

[4]

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0

Question 38

HL • Paper 2

Hard

Calculator Permitted

The displacement $x$ of an unstable system satisfies

\frac{\mathrm d^2x}{\mathrm dt^2}+\frac{\mathrm dx}{\mathrm dt}-6x=0

with $x(0)=1$ and $x'(0)=0$ .

A

I.

Write the equation as a first-order system and classify the equilibrium at the origin.

[3]

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II.

Find expressions for $x(t)$ and $v(t)$ .

[4]

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B

Find the first time after $t=0$ when $x=5$ , and state the asymptote of the trajectory in the $(x,v)$ phase plane as $t\to\infty$ .

[3]

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0

Question 39

HL • Paper 3

Hard

Calculator Permitted

The density $F$ of flowering plants and the density $B$ of pollinating insects are modelled by

\begin{aligned} \frac{dF}{dt}&=0.5F\left(1-\frac{F}{80}\right)-0.04FB,\\ \frac{dB}{dt}&=0.02FB-0.3B. \end{aligned}

Both densities are positive and $t$ is measured in months.

A

Find the positive equilibrium.

I.

Show that the equilibrium value of $F$ is $15$ .

[2]

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II.

Hence find the equilibrium value of $B$ .

[2]

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B

Starting from $(F,B)=(20,8)$ , use one Euler step of length $0.5$ to estimate the two densities.

[2]

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C

pesticide halves the coefficient $0.02$ but leaves all other coefficients unchanged. Determine the new positive equilibrium value of $F$ and interpret the change.

[2]

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Question 40

HL • Paper 3

Hard

Calculator Permitted

Two state variables satisfy

\frac{d\mathbf X}{dt}= \begin{pmatrix}1&2\\2&1\end{pmatrix}\mathbf X, \qquad \mathbf X(0)=\begin{pmatrix}3\\1\end{pmatrix}

A

Find the eigenstructure of the system.

I.

Find the eigenvalues.

[1]

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II.

Find a corresponding eigenvector for each eigenvalue.

[2]

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B

Hence find $x(t)$ and $y(t)$ .

[3]

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C

Describe the trajectory as $t\to\infty$ , including its limiting direction.

[2]

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Question 41

HL • Paper 3

Hard

Calculator Permitted

Temperature deviations $x$ and $y$ in two linked chambers satisfy

\frac{d}{dt}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}-3&1\\1&-3\end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix}, yqquad \begin{pmatrix}x(0)\\y(0)\end{pmatrix}= \begin{pmatrix}6\\2\end{pmatrix}

A

Find the eigenvalues and corresponding eigendirections.

I.

Find the eigenvalues.

[1]

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II.

Find a corresponding eigenvector for each eigenvalue.

[2]

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B

Hence find the particular solution.

[3]

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C

Classify the origin and state the eigendirection approached by this trajectory.

[2]

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Question 42

HL • Paper 3

Hard

Calculator Permitted

A conservative system is modelled by

\frac{dx}{dt}=-2y, \qquad \frac{dy}{dt}=8x

with $(x(0),y(0))=(1,0)$ .

A

Analyse the phase-plane invariant.

I.

Show that $4x^2+y^2$ is constant along every trajectory.

[2]

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II.

Find the equation of the trajectory through $(1,0)$ .

[1]

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B

Find $x(t)$ and $y(t)$ .

[3]

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C

State the period and direction of motion in the phase plane.

[2]

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Question 43

HL • Paper 3

Hard

Calculator Permitted

The vertical displacement $x$ , in metres, of an object released from rest is measured upwards from its release point. Air resistance is proportional to velocity, giving

\frac{d^2x}{dt^2}=-9.8-2\frac{dx}{dt}, \qquad x(0)=0, \qquad x'(0)=0

Here, $t$ is measured in seconds.

A

Use the forward Euler method with step size $h=0.2\ \text{s}$ .

I.

Write the coupled Euler recurrences using $v=x'$ .

[2]

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II.

Estimate $x(0.4)$ and $v(0.4)$ .

[2]

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B

Solve the first-order equation for $v$ and hence find $x(t)$ .

[3]

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C

State the terminal velocity predicted by the model.

[1]

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Question 44

HL • Paper 3

Hard

Calculator Permitted

The angular displacement $\theta$ of a pendulum is modelled by

\frac{d^2\theta}{dt^2}=-4\sin\theta

with $\theta(0)=0.8$ and $\theta'(0)=0$ . Angles are measured in radians.

A

Let $\omega=\theta'$ .

I.

Write the coupled Euler recurrences for step size $h=0.1$ .

[2]

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II.

Estimate $\theta(0.2)$ and $\omega(0.2)$ .

[2]

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B

Show that $E=\dfrac12\omega^2+4(1-\cos\theta)$ is constant for an exact solution.

[3]

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C

Explain why Euler values may not preserve $E$ exactly.

[1]

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Question 45

HL • Paper 3

Hard

Calculator Permitted

A control-system displacement satisfies

\frac{d^2x}{dt^2}-\frac{dx}{dt}-2x=0

Let $v=x'$ .

A

Analyse the corresponding first-order system.

I.

Write the matrix system and find its eigenvalues.

[2]

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II.

Find a corresponding eigenvector for each eigenvalue and classify the origin.

[2]

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B

Given $x(0)=3$ and $v(0)=0$ , find $x(t)$ .

[3]

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C

Deduce the condition relating $x(0)$ and $v(0)$ for a non-zero solution to approach the origin as $t\to\infty$ .

[1]

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Question 46

HL • Paper 3

Hard

Calculator Permitted

The temperature $T$ , in degrees Celsius, inside a greenhouse is modelled by

\frac{dT}{dt}=-0.25\left[T-\left(18+6\sin\left(\frac{\pi t}{12}\right)\right)\right], \qquad T(0)=30

where $t$ is measured in hours.

External temperature over 24 h with T(0)=30.

A

Interpret the rate equation.

I.

Find the curve in the $(t,T)$ plane on which $dT/dt=0$ .

[1]

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II.

State when the greenhouse is warming and when it is cooling.

[2]

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B

Use Euler's method with step size $h=3$ to estimate $T(6)$ .

[3]

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C

numerical solver using a much smaller step gives $T(24)\approx15.0$ . Explain why the greenhouse is warming at $t=24$ despite being cooler than it was initially.

[2]

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Question 47

HL • Paper 2

Hard

Calculator Permitted

The angular displacement $\theta$ , in radians, of a damped pendulum is modelled by

\frac{\mathrm d^2\theta}{\mathrm dt^2}=-0.6\frac{\mathrm d\theta}{\mathrm dt}-4\sin\theta

Initially, $\theta=0.5$ and $\theta'=0$ .

A simple pendulum diagram showing angular displacement $\theta$ from the downward vertical and indicating damping. No computed angles or trajectory are shown.

A

I.

Define $\omega=\theta'$ and write down Euler recurrence relations with step size $h=0.1$ .

[2]

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II.

Use Euler's method to estimate $\theta$ and $\omega$ at $t=0.3$ .

[4]

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B

For small $\theta$ , use $\sin\theta\approx\theta$ to analyse the equilibrium at the origin in the $(\theta,\omega)$ phase plane.

[4]

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Question 48

HL • Paper 3

Hard

Calculator Permitted

A fish population $P$ is subject to constant harvesting. It is modelled by

\frac{dP}{dt}=0.4P\left(1-\frac{P}{1000}\right)-H

where $H$ is the number of fish harvested per year.

A

Suppose that $H=64$ .

I.

Find the two positive equilibrium populations.

[2]

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II.

Determine the stability of each equilibrium.

[2]

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B

Find the equilibrium populations in terms of $H$ .

[2]

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C

Hence determine the greatest harvesting rate for which an equilibrium population exists, and state the corresponding population.

[2]

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0