Question 4 - A-Level Maths

Edexcel FP2 2013 June — Question 4 9 marks

Exam Board	Edexcel
Module	FP2 (Further Pure Mathematics 2)
Year	2013
Session	June
Marks	9
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	First order differential equations (integrating factor)
Type	Standard linear first order - variable coefficients
Difficulty	Standard +0.8 This is a standard integrating factor problem from Further Maths FP2, requiring students to identify the form, calculate the integrating factor (which involves integrating cot x), and perform the integration of sin x times the integrating factor. While the method is standard, the algebraic manipulation and integration steps are non-trivial, placing it moderately above average difficulty.
Spec	4.10c Integrating factor: first order equations

4. Find the general solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + ( 1 + x \cot x ) y = \sin x , \quad 0 < x < \pi$$ giving your answer in the form $y = \mathrm { f } ( x )$.

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Question 4:

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
$\text{IF} = \int\frac{1 + x\cot x}{x}\,dx$	B1
Obtains $\ln x + \ln\sin x$	M1 A1
So $\text{IF} = x\sin x$	A1ft
$\frac{d}{dx}(\text{IF}\cdot y) = \text{IF} \times \frac{\sin x}{x}$	M1
$yx\sin x = \int\sin^2 x\,dx = \int\frac{1-\cos 2x}{2}\,dx = \frac{x}{2} - \frac{1}{4}\sin 2x\ (+c)$	M1 A1
So $y = \frac{1}{2}\text{cosec}\,x - \frac{1}{2x}\cos x + \frac{c}{x\sin x}$	M1 A1
Total	(9)

## Question 4:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\text{IF} = \int\frac{1 + x\cot x}{x}\,dx$ | B1 | |
| Obtains $\ln x + \ln\sin x$ | M1 A1 | |
| So $\text{IF} = x\sin x$ | A1ft | |
| $\frac{d}{dx}(\text{IF}\cdot y) = \text{IF} \times \frac{\sin x}{x}$ | M1 | |
| $yx\sin x = \int\sin^2 x\,dx = \int\frac{1-\cos 2x}{2}\,dx = \frac{x}{2} - \frac{1}{4}\sin 2x\ (+c)$ | M1 A1 | |
| So $y = \frac{1}{2}\text{cosec}\,x - \frac{1}{2x}\cos x + \frac{c}{x\sin x}$ | M1 A1 | |
| **Total** | **(9)** | |

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4. Find the general solution of the differential equation

$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + ( 1 + x \cot x ) y = \sin x , \quad 0 < x < \pi$$

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

\hfill \mbox{\textit{Edexcel FP2 2013 Q4 [9]}}

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Q1 7 Q2 9 Q3 9 Q4 9 Q5 7 Q6 8 Q7 7 Q8 8 Q9 11