Question 3 - A-Level Maths

CAIE Further Paper 2 2020 Specimen — Question 3 8 marks

Exam Board	CAIE
Module	Further Paper 2 (Further Paper 2)
Year	2020
Session	Specimen
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 +0.8 This is a standard first-order linear ODE requiring the integrating factor method, but involves non-trivial integration of trigonometric functions and careful algebraic manipulation. The initial condition application and final simplification add complexity beyond routine textbook exercises, placing it moderately above average difficulty for Further Maths students.
Spec	4.10c Integrating factor: first order equations

3 Find the solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = \frac { \sin x } { x }$$ for which $y = 0$ when $x = \frac { 1 } { 2 } \pi$. Give your answer in the form $y = \mathrm { f } ( x )$.

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Question 3:

Answer	Marks	Guidance
Answer	Marks	Guidance
$\dfrac{dy}{dx} + 3\dfrac{y}{x} = \dfrac{\sin x}{x^2}$	B1	Divide through by $x$
Integrating factor: $e^{\int \frac{3}{x}dx}$	M1	Find integrating factor
$= x^3$	A1
$\dfrac{d}{dx}(yx^3) = x\sin x$	M1	Correct form LHS and attempt to integrate RHS
$yx^3 = -x\cos x + \sin x + c$	A1
$c = -1$	M1	Find $c$
$y = \dfrac{-x\cos x + \sin x - 1}{x^3}$	M1A1	Division through by coefficient of $y$; CAO
8

# Question 3:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{dy}{dx} + 3\dfrac{y}{x} = \dfrac{\sin x}{x^2}$ | B1 | Divide through by $x$ |
| Integrating factor: $e^{\int \frac{3}{x}dx}$ | M1 | Find integrating factor |
| $= x^3$ | A1 | |
| $\dfrac{d}{dx}(yx^3) = x\sin x$ | M1 | Correct form LHS and attempt to integrate RHS |
| $yx^3 = -x\cos x + \sin x + c$ | A1 | |
| $c = -1$ | M1 | Find $c$ |
| $y = \dfrac{-x\cos x + \sin x - 1}{x^3}$ | M1A1 | Division through by coefficient of $y$; CAO |
| | **8** | |

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3 Find the solution of the differential equation

$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = \frac { \sin x } { x }$$

for which $y = 0$ when $x = \frac { 1 } { 2 } \pi$. Give your answer in the form $y = \mathrm { f } ( x )$.\\

\hfill \mbox{\textit{CAIE Further Paper 2 2020 Q3 [8]}}

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Q0 Q1 6 Q2 6 Q3 8 Q4 8 Q5 10 Q6 10 Q7 12 Q8 15