Question 6 - A-Level Maths

Edexcel F2 2020 June — Question 6 8 marks

Exam Board	Edexcel
Module	F2 (Further Pure Mathematics 2)
Year	2020
Session	June
Marks	8
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	First order differential equations (integrating factor)
Type	Standard linear first order - variable coefficients
Difficulty	Challenging +1.2 This is a standard integrating factor problem from Further Maths F2. While it requires dividing by x² to get standard form, recognizing the integrating factor structure, and integrating products involving trigonometric functions, it follows a completely routine method with no novel insight required. The 8 marks reflect mechanical length rather than conceptual difficulty. It's harder than average A-level due to being Further Maths content, but straightforward within that context.
Spec	4.10c Integrating factor: first order equations

6. Obtain the general solution of the equation $$x ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + ( x \cot x + 2 ) x y = 4 \sin x \quad 0 < x < \pi$$ Give your answer in the form $y = \mathrm { f } ( x )$ (8)

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Question 6:

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
$\frac{dy}{dx} + \frac{(x\cot x+2)}{x}y = \frac{4\sin x}{x^2}$	B1	Divide through by $x^2$
$\text{IF} = e^{\int\frac{(x\cot x+2)}{x}dx}$	M1	Attempt IF of form $e^{\int\frac{k(x\cot x+2)}{x}dx}$
$= e^{(\ln\sin x + 2\ln x)}$	A1	$(\ln\sin x + 2\ln x)$
$= x^2\sin x$	A1	Correct IF
$\frac{d}{dx}(\text{IF}\times y) = \text{IF}\times\frac{4\sin x}{x^2}$	M1	Multiply through by IF and write LHS in form shown; allow use of their RHS
$yx^2\sin x = \int 4\sin^2 x\, dx = 4\int\frac{1-\cos 2x}{2}dx = 4\left(\frac{x}{2}-\frac{1}{4}\sin 2x\right)+C$	dM1A1	Attempt to integrate $\sin^2 x$ using $\sin^2 x = \frac{1}{2}(1\pm\cos 2x)$; correct integration, constant not needed
$y = \frac{2x-\sin 2x+C}{x^2\sin x}$	A1cao (8)	Include constant and treat correctly; must have $y=\ldots$

## Question 6:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{dy}{dx} + \frac{(x\cot x+2)}{x}y = \frac{4\sin x}{x^2}$ | B1 | Divide through by $x^2$ |
| $\text{IF} = e^{\int\frac{(x\cot x+2)}{x}dx}$ | M1 | Attempt IF of form $e^{\int\frac{k(x\cot x+2)}{x}dx}$ |
| $= e^{(\ln\sin x + 2\ln x)}$ | A1 | $(\ln\sin x + 2\ln x)$ |
| $= x^2\sin x$ | A1 | Correct IF |
| $\frac{d}{dx}(\text{IF}\times y) = \text{IF}\times\frac{4\sin x}{x^2}$ | M1 | Multiply through by IF and write LHS in form shown; allow use of their RHS |
| $yx^2\sin x = \int 4\sin^2 x\, dx = 4\int\frac{1-\cos 2x}{2}dx = 4\left(\frac{x}{2}-\frac{1}{4}\sin 2x\right)+C$ | dM1A1 | Attempt to integrate $\sin^2 x$ using $\sin^2 x = \frac{1}{2}(1\pm\cos 2x)$; correct integration, constant not needed |
| $y = \frac{2x-\sin 2x+C}{x^2\sin x}$ | A1cao (8) | Include constant and treat correctly; must have $y=\ldots$ |

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6. Obtain the general solution of the equation

$$x ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + ( x \cot x + 2 ) x y = 4 \sin x \quad 0 < x < \pi$$

Give your answer in the form $y = \mathrm { f } ( x )$\\
(8)\\

\hfill \mbox{\textit{Edexcel F2 2020 Q6 [8]}}

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