Further Mathematics · Pearson Edexcel 9FM0 · Option FP1 · Topic 3

Further differential equations

Using Taylor series to find approximate solutions of differential equations, and substitutions that turn harder equations into ones you can solve.

2 sub-topics
Taught in Year 13
19 practice questions
2 animated lessons

Practice questions (19) 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.

3.1Taylor series solutions of differential equationsY13
3.2Reducible differential equationsY13

Try 3 questions

Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 19 questions in this topic, free, no account needed.

Further differential equations

Not needed for Year 12Easy

The method of series solution · Year 13 · 1 mark

How are the higher derivatives at the starting point found for a Taylor series solution of a differential equation?
(1)

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Each differentiation gives the next derivative in terms of the ones before it.A: Differentiate the differential equation again and again, substituting the values already found

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 differential equations

Not needed for Year 12Medium

A series solution · Year 13 · 1 mark

$\dfrac{\mathrm{d}y}{\mathrm{d}x}$ ${}= x + y$ and $y = 2$ when $x = 0$ . Which is the series solution up to the term in $x^2$ ?
(1)

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At $x = 0$ : $y' = 2$ . Differentiating, $y'' = 1 + y' = 3$ . Then $y$ ${}= y_0$ ${}+ y'_0x$ ${}+ \tfrac{1}{2}y''_0x^2$ . $C$ : $2 + 2x + \tfrac{3}{2}x^2$

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 differential equations

Not needed for Year 12Hard

Taylor series solution of a differential equation · Year 13 · 7 marks

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

{}= 2x + y^2

and

y = 1

when

x = 0

.

(a)
Show that $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}$ ${}= 2$ ${}+ 2y\dfrac{\mathrm{d}y}{\mathrm{d}x}$ .
(2)
(b)
Find a series solution for $y$ in ascending powers of $x$ , up to and including the term in $x^3$ .
(5)

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Part (a)2 marks

Implicit differentiation. $\frac{\mathrm{d}}{\mathrm{d}x}(y^2)$ ${}= 2y\frac{\mathrm{d}y}{\mathrm{d}x}$
Differentiate the right side term by term. $\frac{\mathrm{d}^2y}{\mathrm{d}x^2}$ ${}= 2$ ${}+ 2y\frac{\mathrm{d}y}{\mathrm{d}x}$

Part (b)5 marks

$y'(0) = 1$
$y''(0) = 4$
Differentiate again with the product rule. $y'''$ ${}= 2\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2$ ${}+ 2y\frac{\mathrm{d}^2y}{\mathrm{d}x^2}$ ${}\Rightarrow y'''(0)$ ${}= 10$
$y$ ${}= y(0)$ ${}+ y'(0)x$ ${}+ y''(0)x^{2}/2!$ ${}+ y'''(0)x^{3}/3!$ . $y$ ${}= 1$ ${}+ x$ ${}+ 2 x^{2}$ ${}+ \frac{5 x^{3}}{3}$

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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