Question 3 - A-Level Maths

CAIE FP1 2012 November — Question 3 6 marks

Exam Board	CAIE
Module	FP1 (Further Pure Mathematics 1)
Year	2012
Session	November
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Second order differential equations
Type	Standard non-homogeneous with polynomial RHS
Difficulty	Standard +0.8 This is a standard second-order linear differential equation with constant coefficients and polynomial RHS. It requires finding the complementary function (solving the auxiliary equation with complex roots) and a particular integral (trying a quadratic form). While methodical, it involves multiple techniques and careful algebra, making it moderately challenging for Further Maths students but still a routine textbook-style question.
Spec	4.10d Second order homogeneous: auxiliary equation method 4.10e Second order non-homogeneous: complementary + particular integral

3 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 13 x = 26 t ^ { 2 } + 3 t + 13$$

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Question 3:

Answer	Marks	Guidance
Working/Answer	Mark	Guidance
$m^2 + 4m + 13 = 0 \Rightarrow m = -2 \pm 3i$	M1	Solves AQE
CF: $e^{-2t}(A\cos 3t + B\sin 3t)$	A1	Finds CF
PI: $x = pt^2 + qt + r \Rightarrow \dot{x} = 2pt + q \Rightarrow \ddot{x} = 2p$	M1	Form for PI and differentiates
$13p = 26 \Rightarrow p = 2$, $\quad 8p + 13q = 3 \Rightarrow q = -1$, $\quad 2p + 4q + 13r = 13 \Rightarrow r = 1$	M1A1	Compares coefficients and solves
GS: $x = e^{-2t}(A\cos 3t + B\sin 3t) + 2t^2 - t + 1$	A1	Part total 6; Total [6]

## Question 3:

| Working/Answer | Mark | Guidance |
|---|---|---|
| $m^2 + 4m + 13 = 0 \Rightarrow m = -2 \pm 3i$ | M1 | Solves AQE |
| CF: $e^{-2t}(A\cos 3t + B\sin 3t)$ | A1 | Finds CF |
| PI: $x = pt^2 + qt + r \Rightarrow \dot{x} = 2pt + q \Rightarrow \ddot{x} = 2p$ | M1 | Form for PI and differentiates |
| $13p = 26 \Rightarrow p = 2$, $\quad 8p + 13q = 3 \Rightarrow q = -1$, $\quad 2p + 4q + 13r = 13 \Rightarrow r = 1$ | M1A1 | Compares coefficients and solves |
| GS: $x = e^{-2t}(A\cos 3t + B\sin 3t) + 2t^2 - t + 1$ | A1 | Part total 6; **Total [6]** |

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3 Find the general solution of the differential equation

$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 13 x = 26 t ^ { 2 } + 3 t + 13$$

\hfill \mbox{\textit{CAIE FP1 2012 Q3 [6]}}

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Q1 5 Q2 6 Q3 6 Q4 8 Q5 8 Q6 9 Q7 9 Q8 10 Q9 12 Q10 13 Q11 EITHER Q11 OR