CAIE FP1 2013 November — Question 3 7 marks

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2013
SessionNovember
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicSecond order differential equations
TypeStandard non-homogeneous with polynomial RHS
DifficultyStandard +0.8 This is a standard second-order linear non-homogeneous differential equation requiring both complementary function (solving auxiliary equation with complex roots) and particular integral (trying polynomial form). While methodical, it involves multiple techniques and careful algebra, making it moderately challenging for Further Maths students but still a routine textbook exercise.
Spec4.10e Second order non-homogeneous: complementary + particular integral
3 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 4 x ^ { 2 } + 8$$
Question 3:
AnswerMarks Guidance
Working/AnswerMarks Guidance
\(m^2 + 2m + 4 = 0 \Rightarrow m = -1 \pm \sqrt{3}i\)M1 Finds complementary function
C.F. \(e^{-x}(A\cos\sqrt{3}x + B\sin\sqrt{3}x)\)A1
Trial P.I. \(y = px^2 + qx + r\), \(y' = 2px + q\), \(y'' = 2p\)
\(2p + 4px + 2q + 4px^2 + 4qx + 4r = 4x^2 + 8\)M1A1 Differentiates twice and substitutes
\(\Rightarrow p=1, q=-1, r=2\)M1A1 Equates coefficients and solves
\(y = e^{-x}(A\cos\sqrt{3}x + B\sin\sqrt{3}x) + x^2 - x + 2\)A1 (7) States G.S.
Total: [7]
## Question 3:

| Working/Answer | Marks | Guidance |
|---|---|---|
| $m^2 + 2m + 4 = 0 \Rightarrow m = -1 \pm \sqrt{3}i$ | M1 | Finds complementary function |
| C.F. $e^{-x}(A\cos\sqrt{3}x + B\sin\sqrt{3}x)$ | A1 | |
| Trial P.I. $y = px^2 + qx + r$, $y' = 2px + q$, $y'' = 2p$ | | |
| $2p + 4px + 2q + 4px^2 + 4qx + 4r = 4x^2 + 8$ | M1A1 | Differentiates twice and substitutes |
| $\Rightarrow p=1, q=-1, r=2$ | M1A1 | Equates coefficients and solves |
| $y = e^{-x}(A\cos\sqrt{3}x + B\sin\sqrt{3}x) + x^2 - x + 2$ | A1 | **(7)** States G.S. |

**Total: [7]**

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3 Find the general solution of the differential equation

$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 4 x ^ { 2 } + 8$$

\hfill \mbox{\textit{CAIE FP1 2013 Q3 [7]}}
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Q1 6 Q2 6 Q3 7 Q4 7 Q5 8 Q6 8 Q7 9 Q8 11 Q9 11 Q10 13 Q11 EITHER Q11 OR