Question 3 - A-Level Maths

Edexcel FP2 2009 June — Question 3 8 marks

Exam Board	Edexcel
Module	FP2 (Further Pure Mathematics 2)
Year	2009
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	Standard +0.3 This is a standard first-order linear differential equation requiring the integrating factor method. While it's from Further Maths FP2, the technique is routine: divide by sin x to get standard form, identify integrating factor as 1/sin x (or cosec x), multiply through, and integrate. The right-hand side simplifies nicely using sin 2x = 2sin x cos x. It's slightly above average difficulty due to being Further Maths content and requiring careful algebraic manipulation, but follows a well-practiced algorithm without requiring novel insight.
Spec	1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=1 1.05l Double angle formulae: and compound angle formulae 4.10c Integrating factor: first order equations

3. Find the general solution of the differential equation $$\sin x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y \cos x = \sin 2 x \sin x$$ giving your answer in the form $y = \mathrm { f } ( x )$.

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Question 3:

Answer	Marks	Guidance
Divide every term by $\sin x$: $\frac{\mathrm{d}y}{\mathrm{d}x} - \frac{y\cos x}{\sin x} = \sin 2x$	M1	Attempt to divide every term by $\sin x$
Integrating factor $= e^{\int -\frac{\cos x}{\sin x}\,\mathrm{d}x} = e^{-\ln\sin x} = \frac{1}{\sin x}$	dM1	$e^{\int \pm\frac{\cos x}{\sin x}\,\mathrm{d}x}$ or $e^{\int \pm P(x)\,\mathrm{d}x}$
A1 aef	$e^{-\ln\sin x}$ or $e^{\ln\operatorname{cosec}x}$
A1 aef	$\frac{1}{\sin x}$ or $(\sin x)^{-1}$ or $\operatorname{cosec}x$
$\frac{\mathrm{d}}{\mathrm{d}x}\!\left(\frac{y}{\sin x}\right) = \sin 2x \times \frac{1}{\sin x}$	M1	$\frac{\mathrm{d}}{\mathrm{d}x}(y \times \text{I.F.}) = \sin 2x \times \text{I.F.}$
$\frac{\mathrm{d}}{\mathrm{d}x}\!\left(\frac{y}{\sin x}\right) = 2\cos x$	A1	$\frac{\mathrm{d}}{\mathrm{d}x}\!\left(\frac{y}{\sin x}\right)=2\cos x$ or $\frac{y}{\sin x}=\int 2\cos x\,\mathrm{d}x$
$\frac{y}{\sin x} = 2\sin x + K$	dddM1	Credible attempt to integrate RHS with/without $+K$
$y = 2\sin^2 x + K\sin x$	A1 cao	$y = 2\sin^2 x + K\sin x$ (8 marks)

# Question 3:
| Divide every term by $\sin x$: $\frac{\mathrm{d}y}{\mathrm{d}x} - \frac{y\cos x}{\sin x} = \sin 2x$ | M1 | Attempt to divide every term by $\sin x$ |
|---|---|---|
| Integrating factor $= e^{\int -\frac{\cos x}{\sin x}\,\mathrm{d}x} = e^{-\ln\sin x} = \frac{1}{\sin x}$ | dM1 | $e^{\int \pm\frac{\cos x}{\sin x}\,\mathrm{d}x}$ or $e^{\int \pm P(x)\,\mathrm{d}x}$ |
| | A1 **aef** | $e^{-\ln\sin x}$ or $e^{\ln\operatorname{cosec}x}$ |
| | A1 **aef** | $\frac{1}{\sin x}$ or $(\sin x)^{-1}$ or $\operatorname{cosec}x$ |
| $\frac{\mathrm{d}}{\mathrm{d}x}\!\left(\frac{y}{\sin x}\right) = \sin 2x \times \frac{1}{\sin x}$ | M1 | $\frac{\mathrm{d}}{\mathrm{d}x}(y \times \text{I.F.}) = \sin 2x \times \text{I.F.}$ |
| $\frac{\mathrm{d}}{\mathrm{d}x}\!\left(\frac{y}{\sin x}\right) = 2\cos x$ | A1 | $\frac{\mathrm{d}}{\mathrm{d}x}\!\left(\frac{y}{\sin x}\right)=2\cos x$ or $\frac{y}{\sin x}=\int 2\cos x\,\mathrm{d}x$ |
| $\frac{y}{\sin x} = 2\sin x + K$ | dddM1 | Credible attempt to integrate RHS with/without $+K$ |
| $y = 2\sin^2 x + K\sin x$ | A1 **cao** | $y = 2\sin^2 x + K\sin x$ (8 marks) |

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3. Find the general solution of the differential equation

$$\sin x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y \cos x = \sin 2 x \sin x$$

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

\hfill \mbox{\textit{Edexcel FP2 2009 Q3 [8]}}

This paper (8 questions)

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Q1 6 Q2 6 Q3 8 Q4 8 Q5 10 Q6 10 Q7 12 Q8 15

Answer	Marks	Guidance
Divide every term by \(\sin x\): \(\frac{\mathrm{d}y}{\mathrm{d}x} - \frac{y\cos x}{\sin x} = \sin 2x\)	M1	Attempt to divide every term by \(\sin x\)
Integrating factor \(= e^{\int -\frac{\cos x}{\sin x}\,\mathrm{d}x} = e^{-\ln\sin x} = \frac{1}{\sin x}\)	dM1	\(e^{\int \pm\frac{\cos x}{\sin x}\,\mathrm{d}x}\) or \(e^{\int \pm P(x)\,\mathrm{d}x}\)
A1 aef	\(e^{-\ln\sin x}\) or \(e^{\ln\operatorname{cosec}x}\)
A1 aef	\(\frac{1}{\sin x}\) or \((\sin x)^{-1}\) or \(\operatorname{cosec}x\)
\(\frac{\mathrm{d}}{\mathrm{d}x}\!\left(\frac{y}{\sin x}\right) = \sin 2x \times \frac{1}{\sin x}\)	M1	\(\frac{\mathrm{d}}{\mathrm{d}x}(y \times \text{I.F.}) = \sin 2x \times \text{I.F.}\)
\(\frac{\mathrm{d}}{\mathrm{d}x}\!\left(\frac{y}{\sin x}\right) = 2\cos x\)	A1	\(\frac{\mathrm{d}}{\mathrm{d}x}\!\left(\frac{y}{\sin x}\right)=2\cos x\) or \(\frac{y}{\sin x}=\int 2\cos x\,\mathrm{d}x\)
\(\frac{y}{\sin x} = 2\sin x + K\)	dddM1	Credible attempt to integrate RHS with/without \(+K\)
\(y = 2\sin^2 x + K\sin x\)	A1 cao	\(y = 2\sin^2 x + K\sin x\) (8 marks)