Further Mathematics · Pearson Edexcel 9FM0 · Option FP1 · Topic 6

Further numerical methods

Numerical methods for differential equations, such as Euler's method and its improved versions, and Simpson's rule for integrals.

2 sub-topics
1 in Year 12, 1 in Year 13
34 practice questions
2 animated lessons

Practice questions (34) Lessons Worksheets All Edexcel Further Mathematics topics

What the specification says

Every sub-topic Pearson Edexcel can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

6.1Numerical solution of differential equationsY12
6.2Simpson's ruleY13

Try 3 questions

Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 34 questions in this topic, free, no account needed.

Further numerical methods

Easy

One step of Euler's method · Year 12 · 1 mark

$\dfrac{\mathrm{d}y}{\mathrm{d}x}$ ${}= xy$ and $y = 2$ when $x = 1$ . Using Euler's method with step length 0.2, what is the estimate of $y$ at $x = 1.2$ ?
(1)

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$y_1 = y_0 + h\,f(x_0, y_0)$ .A: 2.4

Total1

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Further numerical methods

Not needed for Year 12Medium

Simpson's rule · Year 13 · 7 marks

I

{}= \displaystyle\int_{1}^{2} \frac{1}{x}\,\mathrm{d}x

.

(a)
Use Simpson's rule with 6 strips to estimate I, giving your answer to 5 decimal places.
(4)
(b)
Find the exact value of I, and hence the percentage error in your estimate.
(3)

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Part (a)4 marks

6 strips need 7 ordinates. $h = \frac{1}{6}$
Work out every ordinate. $1.00000, \ 0.85714, \ 0.75000, \ 0.66667, \ 0.60000, \ 0.54545, \ 0.50000$
Simpson's rule. $\tfrac{h}{3}\left[y_0 + y_n + 4(\text{odd}) + 2(\text{even})\right]$
$I \approx 0.69317$

Part (b)3 marks

$I = \ln 2$
$\frac{|0.69317 - 0.69315|}{0.69315} \times 100$
$= 0.0033\%$

Total7

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Further numerical methods

Hard

Numerical solution of a differential equation · Year 12 · 6 marks

\dfrac{\mathrm{d}y}{\mathrm{d}x}

{}= 2x + \frac{y^2}{2}

, with

y = 1

at

x = 0

.

(a)
Use Euler's method with step length 0.1 to find an estimate of $y$ at $x = 0.1$ .
(2)
(b)
Use the approximation $\displaystyle y_{r + 1}$ ${}\approx y_{r - 1}$ ${}+ 2h\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)_r$ with your answer to (a) to estimate $y$ at $x = 0.2$ and at $x = 0.3$ , giving answers to 3 decimal places.
(4)

Check your answer before you look at the solution

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Part (a)2 marks

$y_1$ ${}= y_0$ ${}+ h\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)_0$ ${}= 1 + 0.1 \times 0.5$
$y_1 = 1.05$

Part (b)4 marks

Use y₁ from (a). $\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)_1$ ${}= 0.7512$
Keep the unrounded value for the next step. $y_2$ ${}= 1 + 0.2 \times 0.7512$ ${}= 1.150$
$\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)_2$ ${}= 1.0615$
$y_3$ ${}= 1.05 + 0.2 \times 1.0615$ ${}= 1.262$

Total6

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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