AQA FP3 2007 January — Question 5 12 marks

Exam BoardAQA
ModuleFP3 (Further Pure Mathematics 3)
Year2007
SessionJanuary
Marks12
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Mark schemeDownload PDF ↗
TopicSecond order differential equations
TypeCombined polynomial and trigonometric RHS
DifficultyChallenging +1.2 This is a standard second-order linear differential equation with constant coefficients from Further Maths FP3. While it requires multiple steps (finding complementary function via auxiliary equation, then particular integral for both polynomial and trigonometric terms), the method is entirely procedural with no novel insight needed. The 12 marks reflect the length rather than conceptual difficulty. It's harder than average A-level due to being Further Maths content, but routine within that context.
Spec4.10e Second order non-homogeneous: complementary + particular integral
5 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = 6 + 5 \sin x$$ (12 marks)
Question 5:
AnswerMarks Guidance
WorkingMarks Guidance
Aux. eqn \(m^2 - 4m + 3 = 0\)M1 PI
\(m = 3\) and \(1\)A1 PI
CF is \(Ae^{3x} + Be^x\)A1F
PI Try \(y = a + b\sin x + c\cos x\)M1 Condone '\(a\)' missing here
\(y'(x) = b\cos x - c\sin x\)A1
\(y''(x) = -b\sin x - c\cos x\)A1F ft can be consistent sign error(s)
Substitute into DE givesM1
\(a = 2\)B1
\(4c + 2b = 5\) and \(2c - 4b = 0\)A1
\(b = 0.5\)A1F ft a slip
\(c = 1\)A1F ft a slip
GS: \(y = Ae^{3x} + Be^x + 2 + 0.5\sin x + \cos x\)B1F Total: 12 
## Question 5:

| Working | Marks | Guidance |
|---------|-------|----------|
| Aux. eqn $m^2 - 4m + 3 = 0$ | M1 | PI |
| $m = 3$ and $1$ | A1 | PI |
| CF is $Ae^{3x} + Be^x$ | A1F | |
| PI Try $y = a + b\sin x + c\cos x$ | M1 | Condone '$a$' missing here |
| $y'(x) = b\cos x - c\sin x$ | A1 | |
| $y''(x) = -b\sin x - c\cos x$ | A1F | ft can be consistent sign error(s) |
| Substitute into DE gives | M1 | |
| $a = 2$ | B1 | |
| $4c + 2b = 5$ and $2c - 4b = 0$ | A1 | |
| $b = 0.5$ | A1F | ft a slip |
| $c = 1$ | A1F | ft a slip |
| GS: $y = Ae^{3x} + Be^x + 2 + 0.5\sin x + \cos x$ | B1F | Total: 12 | $y =$ candidate's CF and candidate's PI (must have exactly two arbitrary constants) |

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5 Find the general solution of the differential equation

$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = 6 + 5 \sin x$$

(12 marks)

\hfill \mbox{\textit{AQA FP3 2007 Q5 [12]}}
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