IB Mathematics Applications & Interpretation Topic AHL 5.14—setting Up a DE, Solve by Separating Variables Questionbank (HL)

Question 1

HLPaper 2

A predator-prey system models the populations of rabbits ( $R$ , in thousands) and foxes ( $F$ , in hundreds) in a forest, given by:

$\frac{d R}{d t} = 0.4 R - 0.02 R F, \frac{d F}{d t} = 0.01 R F - 0.3 F$

where $t$ is in years. Initially, $R = 10, F = 5$ .

1.

Find the non-zero equilibrium point for the system.

[3]

2.

Interpret the equilibrium point in the context of the model.

[1]

3.

Use Euler's method with a step size of 0.2 years to estimate $R$ and $F$ after $0.2$ years.

[2]

4.

On a phase plane with $R$ on the horizontal axis and $F$ on the vertical axis, for $8 \leq R \leq 20$ and $3 \leq F \leq 6$ , sketch the initial direction of the trajectory from $(10, 5)$ .

[2]

Question 2

HLPaper 2

A patient receives a drug by intravenous infusion at a constant rate of $18 mg$ per hour. At the same time, the drug is eliminated from the bloodstream at a rate proportional to the amount $D (t)$ present at time $t$ hours, with elimination constant $k > 0$ (in $h^{- 1}$ ).

1.

Formulate a differential equation for $\frac{d D}{d t}$ in terms of $D$ and $k$ .

[2]

2.

Solve the differential equation to find an expression for $D$ in terms of $t$ , $k$ , and a constant of integration $A$ .

[6]

3.

Given that the initial amount of the drug is $6 mg$ , determine the value of $A$ and state the particular solution for $D (t)$ in terms of $k$ and $t$ .

[4]

4.

Find the steady-state amount of drug that will eventually be reached as $t$ becomes very large.

[3]

Question 3

HLPaper 2

An object is released from rest at the top of a long vertical tube filled with a thick viscous fluid at time $t = 0$ seconds. Take $x (0) = 0$ and $v (0) = 0$ , where $x$ is the object’s displacement from the top of the tube measured in metres, and $v = \frac{d x}{d t}$ is its velocity in $m s^{- 1}$ .

Initially, the resistance is modelled as proportional to the velocity, giving the differential equation

$\frac{d ^{2} x}{d t ^{2}} = 9.81 - 0.9 (\frac{d x}{d t})$

where $9.81$ is in $m s^{- 2}$ and the constant $0.9$ has units $s^{- 1}$ .

The maximum velocity approached by the object as it falls is known as the terminal velocity.

An experiment is performed in which the object is placed in the fluid on a number of occasions and its terminal velocity is recorded. It is found that the terminal velocity was consistently smaller than that predicted by the model used. It was suggested that the resistance to motion is actually proportional to the velocity squared and so the following model was set up:

$\frac{d ^{2} x}{d t ^{2}} = 9.81 - 0.9 (\frac{d x}{d t})^{2}$

In this second model, the constant $0.9$ has units $m^{- 1}$ .

At terminal velocity, the acceleration of an object is equal to zero.

1.

By substituting $v = \frac{d x}{d t}$ into the equation, find an expression for the velocity of the object at time $t$ . Give your answer in the form $v = f (t)$ .

[7]

2.

From your solution to part 1, or otherwise, find the terminal velocity of the object predicted by this model.

[2]

3.

Write down the differential equation for the second model as a system of first-order differential equations.

[2]

4.

Use Euler’s method, with a step length of $0.2 s$ , to find the displacement and velocity of the object when $t = 0.6$ .

[4]

5.

By repeated application of Euler’s method, state an approximation for the terminal velocity, to five significant figures.

[1]

6.

Use the differential equation to find the exact terminal velocity for the object predicted by the second model.

[2]

7.

Use your answers to parts 4, 5 and 6 to comment on the accuracy of the Euler approximation to this model.

[2]

Question 4

HLPaper 2

The speed $v (t)$ (in $m s^{- 1}$ ) of a skydiver $t$ seconds after a parachute opens is modelled by $\frac{d v}{d t} = 0.6 (12 - v)$ , $v (0) = 30$ .

1.

Calculate $v (3)$ using Euler's method with a step size $h = 1.5$ .

[4]

2.

Solve the differential equation analytically to find the exact solution for $v (t)$ .

[4]

3.

Determine the percentage error in your Euler approximation.

[2]

4.

State the limiting speed as $t \to \infty$ .

[1]

Question 5

HLPaper 2

A medicine in the bloodstream is eliminated according to the differential equation

$\frac{d M}{d t} = - 0.04 M^{2}$

where $M$ is the mass of medicine (in $mg$ ) and $t$ is time in hours. Initially, at $t = 0$ , $M = 5$ .

1.

Solve the differential equation to find $M$ in terms of $t$ .

[5]

2.

Find the time when the mass of medicine is $1 mg$ .

[3]

3.

Determine the rate of change of the mass when $M = 2 mg$ .

[2]

IB Mathematics Applications & Interpretation AHL 5.14—setting Up a DE, Solve by Separating Variables Questionbank

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IB Mathematics Applications & Interpretation AHL 5.14—setting Up a DE, Solve by Separating Variables Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 5.14—setting Up a DE, Solve by Separating Variables Questionbank

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