Question 5 - A-Level Maths

Edexcel FP2 2005 June — Question 5 7 marks

Exam Board	Edexcel
Module	FP2 (Further Pure Mathematics 2)
Year	2005
Session	June
Marks	7
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	First order differential equations (integrating factor)
Type	Standard linear first order - variable coefficients
Difficulty	Standard +0.8 This is a standard integrating factor problem from Further Maths FP2, requiring division by (x+1) to get standard form, finding integrating factor (x+1)², and integrating 1/[x(x+1)] using partial fractions. While methodical, it involves multiple techniques and careful algebra, placing it moderately above average difficulty for A-level but routine for Further Maths students.
Spec	4.10c Integrating factor: first order equations

5. Find the general solution of the differential equation $$( x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y = \frac { 1 } { x } , \quad x > 0 .$$ giving your answer in the form $y = \mathrm { f } ( x )$.
(7)(Total 7 marks)

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Question 5:

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
$\frac{dy}{dx} + \frac{2}{1+x}y = \frac{1}{x(x+1)}$	M1	Attempt $y' + Py = Q$ form
I.F. $= e^{\int\frac{2}{1+x}dx} = e^{2\ln(1+x)} = (1+x)^2$	M1, A1
$y(1+x)^2 = \int\left(\frac{x+1}{x}\right)dx$ OR $\frac{d}{dx}(y(1+x)^2) = \frac{x+1}{x}$	M1	ft I.F.
i.e. $y(1+x)^2 = x + \ln x + C$	M1 A1
$y = \frac{x + \ln x + C}{(1+x)^2}$	A1	c.a.o.; 7 marks total

# Question 5:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{dy}{dx} + \frac{2}{1+x}y = \frac{1}{x(x+1)}$ | M1 | Attempt $y' + Py = Q$ form |
| I.F. $= e^{\int\frac{2}{1+x}dx} = e^{2\ln(1+x)} = (1+x)^2$ | M1, A1 | |
| $y(1+x)^2 = \int\left(\frac{x+1}{x}\right)dx$ OR $\frac{d}{dx}(y(1+x)^2) = \frac{x+1}{x}$ | M1 | ft I.F. |
| i.e. $y(1+x)^2 = x + \ln x + C$ | M1 A1 | |
| $y = \frac{x + \ln x + C}{(1+x)^2}$ | A1 | c.a.o.; 7 marks total |

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5. Find the general solution of the differential equation

$$( x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y = \frac { 1 } { x } , \quad x > 0 .$$

giving your answer in the form $y = \mathrm { f } ( x )$.\\
(7)(Total 7 marks)\\

\hfill \mbox{\textit{Edexcel FP2 2005 Q5 [7]}}

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Q1 5 Q2 7 Q3 12 Q4 13 Q5 7 Q6 12 Q7 14 Q8 13 Q9 11 Q10 12 Q11 14