Question 5 - A-Level Maths

CAIE Further Paper 2 2021 November — Question 5 11 marks

Exam Board	CAIE
Module	Further Paper 2 (Further Paper 2)
Year	2021
Session	November
Marks	11
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Second order differential equations
Type	Particular solution with initial conditions
Difficulty	Standard +0.8 This is a second-order linear ODE requiring complementary function (repeated root λ=1), particular integral using undetermined coefficients for cos x (involving both sin x and cos x terms), then applying two initial conditions to find constants. While methodical, it requires multiple techniques and careful algebra, making it moderately challenging but still a standard Further Maths question.
Spec	4.10e Second order non-homogeneous: complementary + particular integral

5 Find the particular solution of the differential equation $$\frac { d ^ { 2 } y } { d x ^ { 2 } } - 2 \frac { d y } { d x } + y = 4 \cos x$$ given that, when $x = 0 , y = - 4$ and $\frac { d y } { d x } = 3$.

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Question 5:

Answer	Marks	Guidance
Answer	Marks	Guidance
$m^2 - 2m + 1 = 0 \Rightarrow m = 1$	M1	Auxiliary equation
$y = e^x(Ax + B)$	A1	Complementary function. Accept '$y=$' missing
$y = p\sin x + q\cos x$, $y' = p\cos x - q\sin x$, $y'' = -p\sin x - q\cos x$	M1 A1	Particular integral and its derivatives
$-p + 2q + p = 0 \quad -q - 2p + q = 4$	M1	Substitutes and equates coefficients
$p = -2 \quad q = 0$	A1
$y = e^x(Ax + B) - 2\sin x$	A1
$y' = Ae^x + e^x(Ax + B) - 2\cos x$	M1	Differentiates
$B = -4 \quad A + B - 2 = 3 \Rightarrow A = 9$	M1 A1	Forms simultaneous equations using initial conditions
$y = e^x(9x - 4) - 2\sin x$	A1
11

## Question 5:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $m^2 - 2m + 1 = 0 \Rightarrow m = 1$ | M1 | Auxiliary equation |
| $y = e^x(Ax + B)$ | A1 | Complementary function. Accept '$y=$' missing |
| $y = p\sin x + q\cos x$, $y' = p\cos x - q\sin x$, $y'' = -p\sin x - q\cos x$ | M1 A1 | Particular integral and its derivatives |
| $-p + 2q + p = 0 \quad -q - 2p + q = 4$ | M1 | Substitutes and equates coefficients |
| $p = -2 \quad q = 0$ | A1 | |
| $y = e^x(Ax + B) - 2\sin x$ | A1 | |
| $y' = Ae^x + e^x(Ax + B) - 2\cos x$ | M1 | Differentiates |
| $B = -4 \quad A + B - 2 = 3 \Rightarrow A = 9$ | M1 A1 | Forms simultaneous equations using initial conditions |
| $y = e^x(9x - 4) - 2\sin x$ | A1 | |
| | **11** | |

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5 Find the particular solution of the differential equation

$$\frac { d ^ { 2 } y } { d x ^ { 2 } } - 2 \frac { d y } { d x } + y = 4 \cos x$$

given that, when $x = 0 , y = - 4$ and $\frac { d y } { d x } = 3$.\\

\hfill \mbox{\textit{CAIE Further Paper 2 2021 Q5 [11]}}

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Q1 5 Q2 6 Q3 8 Q4 10 Q5 11 Q6 10 Q7 11 Q8 14