Question 2 - A-Level Maths

CAIE FP1 2017 November — Question 2 6 marks

Exam Board	CAIE
Module	FP1 (Further Pure Mathematics 1)
Year	2017
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 non-homogeneous differential equation requiring both complementary function (solving auxiliary equation with complex roots) and particular integral (trying polynomial form with undetermined coefficients). While the method is systematic, it involves multiple steps including complex number manipulation, differentiation of trial solutions, and coefficient matching, making it moderately challenging for Further Maths students but still a textbook-style question.
Spec	4.10d Second order homogeneous: auxiliary equation method 4.10e Second order non-homogeneous: complementary + particular integral

2 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 4 - 5 t ^ { 2 }$$

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Question 2:

Answer	Marks	Guidance
Answer	Marks	Guidance
CF: $m^2 + 2m + 5 = 0 \Rightarrow m = -1 \pm 2i$	M1
$e^{-t}(A\cos 2t + B\sin 2t)$	A1
PI: $x = pt^2 + qt + r \Rightarrow \dot{x} = 2pt + q \Rightarrow \ddot{x} = 2p$	M1
$2p + 4pt + 2q + 5pt^2 + 5qt + 5r = 4 - 5t^2$	M1
$\Rightarrow p = -1,\ q = \frac{4}{5},\ r = \frac{22}{25}$	A1
GS: $x = e^{-t}(A\cos 2t + B\sin 2t) + \frac{22}{25} + \frac{4}{5}t - t^2$	A1FT
Total	6

## Question 2:

| Answer | Marks | Guidance |
|--------|-------|----------|
| CF: $m^2 + 2m + 5 = 0 \Rightarrow m = -1 \pm 2i$ | M1 | |
| $e^{-t}(A\cos 2t + B\sin 2t)$ | A1 | |
| PI: $x = pt^2 + qt + r \Rightarrow \dot{x} = 2pt + q \Rightarrow \ddot{x} = 2p$ | M1 | |
| $2p + 4pt + 2q + 5pt^2 + 5qt + 5r = 4 - 5t^2$ | M1 | |
| $\Rightarrow p = -1,\ q = \frac{4}{5},\ r = \frac{22}{25}$ | A1 | |
| GS: $x = e^{-t}(A\cos 2t + B\sin 2t) + \frac{22}{25} + \frac{4}{5}t - t^2$ | A1FT | |
| **Total** | **6** | |

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2 Find the general solution of the differential equation

$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 4 - 5 t ^ { 2 }$$

\hfill \mbox{\textit{CAIE FP1 2017 Q2 [6]}}

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Q1 4 Q2 6 Q3 7 Q4 8 Q5 8 Q6 9 Q7 10 Q8 11 Q9 12 Q10 12 Q11 EITHER Q11 OR