Question 4 - A-Level Maths

CAIE Further Paper 2 2020 November — Question 4 8 marks

Exam Board	CAIE
Module	Further Paper 2 (Further Paper 2)
Year	2020
Session	November
Marks	8
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	First order differential equations (integrating factor)
Type	Standard linear first order - variable coefficients
Difficulty	Standard +0.3 This is a standard integrating factor question with straightforward algebra. The integrating factor is x², the integration of e^x/x² requires integration by parts (twice), and applying the initial condition is routine. While it requires multiple techniques, it follows a well-practiced method with no conceptual surprises, making it slightly easier than average.
Spec	4.10c Integrating factor: first order equations

4 Find the solution of the differential equation $$x \frac { d y } { d x } + 2 y = e ^ { x }$$ for which $y = 3$ when $x = 1$. Give your answer in the form $y = f ( x )$.

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Question 4:

Answer	Marks	Guidance
$\frac{dy}{dx} + \frac{2}{x}y = \frac{1}{x}e^x$	B1	Divides through by $x$
$e^{2\int x^{-1}dx} = x^2$	M1 A1	Finds integrating factor, must be integrating $x^{-1}$
$\frac{d}{dx}(yx^2) = xe^x$	M1	Correct form on LHS and attempt to integrate $\frac{1}{x}e^x$ multiplied by their integrating factor
$yx^2 = (x-1)e^x + C$	A1
$C = 3$	M1	Finds $C$
$y = \frac{(x-1)e^x + 3}{x^2}$	M1 A1	Divides through by coefficient of $y$

## Question 4:

| $\frac{dy}{dx} + \frac{2}{x}y = \frac{1}{x}e^x$ | B1 | Divides through by $x$ |
|---|---|---|
| $e^{2\int x^{-1}dx} = x^2$ | M1 A1 | Finds integrating factor, must be integrating $x^{-1}$ |
| $\frac{d}{dx}(yx^2) = xe^x$ | M1 | Correct form on LHS and attempt to integrate $\frac{1}{x}e^x$ multiplied by their integrating factor |
| $yx^2 = (x-1)e^x + C$ | A1 | |
| $C = 3$ | M1 | Finds $C$ |
| $y = \frac{(x-1)e^x + 3}{x^2}$ | M1 A1 | Divides through by coefficient of $y$ |

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4 Find the solution of the differential equation

$$x \frac { d y } { d x } + 2 y = e ^ { x }$$

for which $y = 3$ when $x = 1$. Give your answer in the form $y = f ( x )$.\\

\hfill \mbox{\textit{CAIE Further Paper 2 2020 Q4 [8]}}

This paper (9 questions)

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Q1 5 Q2 6 Q3 4 Q4 8 Q5 8 Q6 11 Q7 7 Q8 10 Q9 16

Answer	Marks	Guidance
\(\frac{dy}{dx} + \frac{2}{x}y = \frac{1}{x}e^x\)	B1	Divides through by \(x\)
\(e^{2\int x^{-1}dx} = x^2\)	M1 A1	Finds integrating factor, must be integrating \(x^{-1}\)
\(\frac{d}{dx}(yx^2) = xe^x\)	M1	Correct form on LHS and attempt to integrate \(\frac{1}{x}e^x\) multiplied by their integrating factor
\(yx^2 = (x-1)e^x + C\)	A1
\(C = 3\)	M1	Finds \(C\)
\(y = \frac{(x-1)e^x + 3}{x^2}\)	M1 A1	Divides through by coefficient of \(y\)