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Date	November Example question	Marks available	4	Reference code	EXN.1.AHL.TZ0.13
Level	Additional Higher Level	Paper	Paper 1	Time zone	Time zone 0
Command term	Find and Estimate	Question number	13	Adapted from	N/A

Question

Consider the second order differential equation

$\ddot{x} + 4 {(\dot{x})}^{2} - 2 t = 0$

where $x$ is the displacement of a particle for $t \geq 0$ .

Write the differential equation as a system of coupled first order differential equations.

[2]

When $t = 0$ , $x = \dot{x} = 0$

Use Euler’s method with a step length of $0.1$ to find an estimate for the value of the displacement and velocity of the particle when $t = 1$ .

[4]

Markscheme

* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.

$\dot{x} = y$ M1

$\dot{y} = 2 t - 4 y^{2}$ A1

[2 marks]

$t_{n + 1} = t_{n} + 0.1$

$x_{n + 1} = x_{n} + 0.1 y_{n}$

$y_{n + 1} = y_{n} + 0.1 (2 t_{n} - 4 y_{n}^{2})$ (M1)(A1)

Note: Award M1 for a correct attempt to substitute the functions in part (a) into the formula for Euler’s method for coupled systems.

When $t = 1$

$x = 0.202 (0.20201 \dots)$ A1

$\dot{x} = 0.598 (0.59822 \dots)$ A1

Note: Accept $y = 0.598$ .

[4 marks]

Examiners report

[N/A]

Syllabus sections

Topic 5—Calculus » AHL 5.18—Eulers method for 2nd order DEs

22M.2.AHL.TZ1.4b.iii:
Find the long-term velocity of the particle.
22M.2.AHL.TZ1.4b.ii:
Use Euler’s method with a step length of $0.1$ to find the displacement of the particle when $t = 1$ .
22M.2.AHL.TZ1.4b.i:
Use the substitution $y = \frac{d x}{d t}$ to write the differential equation as a system of coupled, first order differential equations.
SPM.2.AHL.TZ0.7e:
By repeated application of Euler’s method, find an approximation for the terminal velocity, to five significant figures.
SPM.2.AHL.TZ0.7g:
Use your answers to parts (d), (e) and (f) to comment on the accuracy of the Euler approximation to this model.
SPM.2.AHL.TZ0.7c:
Write down the differential equation as a system of first order differential equations.
SPM.2.AHL.TZ0.7f:
Use the differential equation to find the terminal velocity for the object.
SPM.2.AHL.TZ0.7b:
From your solution to part (a), or otherwise, find the terminal velocity of the object predicted by this model.
SPM.2.AHL.TZ0.7a:
By substituting $v = \frac{{{\text{d}}x}}{{{\text{d}}t}}$ into the equation, find an expression for the velocity of the particle at time $t$ . Give your answer in the form $v = f(t)$ .
EXN.1.AHL.TZ0.13a:
Write the differential equation as a system of coupled first order differential equations.
SPM.2.AHL.TZ0.7d:
Use Euler’s method, with a step length of 0.2, to find the displacement and velocity of the object when $t = 0.6$ .

Topic 5—Calculus

Question

Markscheme

Examiners report

Syllabus sections

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