AQA FP3 2010 January — Question 7 8 marks

Exam BoardAQA
ModuleFP3 (Further Pure Mathematics 3)
Year2010
SessionJanuary
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicSecond order differential equations
TypeCombined polynomial and trigonometric RHS
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 (using two separate trial functions for polynomial and trigonometric terms). While methodical, it requires multiple techniques, careful algebra with the sin x term (where ω matches a CF frequency requiring amplitude adjustment), and synthesis of results—moderately above average for Further Maths.
Spec4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral
7 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = 8 x ^ { 2 } + 9 \sin x$$ (8 marks)
AnswerMarks Guidance
Aux. eqn. \(m^2 + 4 = 0 \Rightarrow m = \pm 2i\)B1
CF is \(A\cos 2x + B\sin 2x\)M1, A1F OE. If m is real give M0; ft on incorrect complex value for m
PI: Try \(ax^2 + b + c\sin x\)M1 Award even if extra terms, provided the relevant coefficients are shown to be zero.
M1
\(2a - c\sin x + 4ax^2 + 4b + 4c\sin x = 8x^2 + 9\sin x\)
AnswerMarks Guidance
\(a = 2, b = -1,\)A1 Dep on relevant M mark
\(c = 3\)A1 Dep on relevant M mark
\((y =) A\cos 2x + B\sin 2x + 2x^2 - 1 + 3\sin x\)B1F Their CF + their PI. Must be exactly two arbitrary constants; 8 marks total
Total: 8 marks
Aux. eqn. $m^2 + 4 = 0 \Rightarrow m = \pm 2i$ | B1 |

CF is $A\cos 2x + B\sin 2x$ | M1, A1F | OE. If m is real give M0; ft on incorrect complex value for m

PI: Try $ax^2 + b + c\sin x$ | M1 | Award even if extra terms, provided the relevant coefficients are shown to be zero.
| M1 |

$2a - c\sin x + 4ax^2 + 4b + 4c\sin x = 8x^2 + 9\sin x$

$a = 2, b = -1,$ | A1 | Dep on relevant M mark

$c = 3$ | A1 | Dep on relevant M mark

$(y =) A\cos 2x + B\sin 2x + 2x^2 - 1 + 3\sin x$ | B1F | Their CF + their PI. Must be exactly two arbitrary constants; 8 marks total

**Total: 8 marks**

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7 Find the general solution of the differential equation

$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = 8 x ^ { 2 } + 9 \sin x$$

(8 marks)

\hfill \mbox{\textit{AQA FP3 2010 Q7 [8]}}
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Q1 8 Q2 8 Q3 9 Q4 5 Q5 12 Q6 9 Q7 8 Q8 16