CAIE Further Paper 2 2023 November — Question 4 10 marks

Exam BoardCAIE
ModuleFurther Paper 2 (Further Paper 2)
Year2023
SessionNovember
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicSecond order differential equations
TypeParticular solution with initial conditions
DifficultyStandard +0.8 This is a standard second-order linear differential equation with constant coefficients requiring both complementary function (solving auxiliary equation with complex roots) and particular integral (trying polynomial form), followed by applying two initial conditions. While methodical, it involves multiple techniques and careful algebra across several steps, making it moderately challenging but within expected Further Maths scope.
Spec4.10e Second order non-homogeneous: complementary + particular integral
4 Find the particular solution of the differential equation $$\frac { d ^ { 2 } y } { d x ^ { 2 } } + 2 \frac { d y } { d x } + 3 y = 27 x ^ { 2 }$$ given that, when \(x = 0 , y = 2\) and \(\frac { \mathrm { dy } } { \mathrm { dx } } = - 8\).
Question 4:
AnswerMarks Guidance
AnswerMarks Guidance
\(m^2 + 2m + 3 = 0\)M1 Auxiliary equation
\([y=]e^{-x}\left(A\cos\sqrt{2}x + B\sin\sqrt{2}x\right)\)A1 Complementary function. Allow with "\(y=\)" missing
\(y = px^2 + qx + r \Rightarrow y' = 2px + q \Rightarrow y'' = 2p\)B1 Particular integral and its derivatives
\(3p = 27\), \(3q + 4p = 0\), \(2p + 2q + 3r = 0\)M1 Substitutes and equates coefficients
\(p = 9\), \(q = -12\), \(r = 2\)A1
\(y = e^{-x}\left(A\cos\sqrt{2}x + B\sin\sqrt{2}x\right) + 9x^2 - 12x + 2\)A1 General solution. Must have "\(y=\)"
\(y' = e^{-x}\left(-\sqrt{2}A\sin\sqrt{2}x + \sqrt{2}B\cos\sqrt{2}x\right) - e^{-x}\left(A\cos\sqrt{2}x + B\sin\sqrt{2}x\right) + 18x - 12\)M1* Differentiates. Must use product rule
\(A + 2 = 2\), \(\sqrt{2}B - A - 12 = -8 \Rightarrow A = 0,\ B = 2\sqrt{2}\)DM1 A1 Uses initial conditions
\(y = 2\sqrt{2}e^{-x}\sin\sqrt{2}x + 9x^2 - 12x + 2\)A1 Must have "\(y=\)"
10 
## Question 4:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $m^2 + 2m + 3 = 0$ | M1 | Auxiliary equation |
| $[y=]e^{-x}\left(A\cos\sqrt{2}x + B\sin\sqrt{2}x\right)$ | A1 | Complementary function. Allow with "$y=$" missing |
| $y = px^2 + qx + r \Rightarrow y' = 2px + q \Rightarrow y'' = 2p$ | B1 | Particular integral and its derivatives |
| $3p = 27$, $3q + 4p = 0$, $2p + 2q + 3r = 0$ | M1 | Substitutes and equates coefficients |
| $p = 9$, $q = -12$, $r = 2$ | A1 | |
| $y = e^{-x}\left(A\cos\sqrt{2}x + B\sin\sqrt{2}x\right) + 9x^2 - 12x + 2$ | A1 | General solution. Must have "$y=$" |
| $y' = e^{-x}\left(-\sqrt{2}A\sin\sqrt{2}x + \sqrt{2}B\cos\sqrt{2}x\right) - e^{-x}\left(A\cos\sqrt{2}x + B\sin\sqrt{2}x\right) + 18x - 12$ | M1* | Differentiates. Must use product rule |
| $A + 2 = 2$, $\sqrt{2}B - A - 12 = -8 \Rightarrow A = 0,\ B = 2\sqrt{2}$ | DM1 A1 | Uses initial conditions |
| $y = 2\sqrt{2}e^{-x}\sin\sqrt{2}x + 9x^2 - 12x + 2$ | A1 | Must have "$y=$" |
| | **10** | |

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4 Find the particular solution of the differential equation

$$\frac { d ^ { 2 } y } { d x ^ { 2 } } + 2 \frac { d y } { d x } + 3 y = 27 x ^ { 2 }$$

given that, when $x = 0 , y = 2$ and $\frac { \mathrm { dy } } { \mathrm { dx } } = - 8$.\\

\hfill \mbox{\textit{CAIE Further Paper 2 2023 Q4 [10]}}
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Q1 4 Q2 5 Q3 6 Q4 10 Q5 10 Q6 14 Q7 11 Q8 15