Question 5 - A-Level Maths

CAIE P3 2010 June — Question 5 6 marks

Exam Board	CAIE
Module	P3 (Pure Mathematics 3)
Year	2010
Session	June
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	First order differential equations (integrating factor)
Type	Separable variables
Difficulty	Standard +0.3 This is a separable differential equation requiring standard separation of variables technique, integration of rational functions, and application of an initial condition. While it involves multiple steps (separation, integration, applying boundary condition, rearranging for y²), each step follows routine procedures without requiring novel insight or particularly challenging integration.
Spec	1.08k Separable differential equations: dy/dx = f(x)g(y)

5 Given that $y = 0$ when $x = 1$, solve the differential equation $$x y \frac { \mathrm {~d} y } { \mathrm {~d} x } = y ^ { 2 } + 4 ,$$ obtaining an expression for $y ^ { 2 }$ in terms of $x$.

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Question 5:

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
Separate variables correctly	B1
Integrate and obtain term $\ln x$	B1
Integrate and obtain term $\frac{1}{2}\ln(y^2 + 4)$	B1
Evaluate a constant or use limits $y = 0$, $x = 1$ in a solution containing $a\ln x$ and $b\ln(y^2+4)$	M1
Obtain correct solution in any form, e.g. $\frac{1}{2}\ln(y^2+4) = \ln x + \frac{1}{2}\ln 4$	A1
Rearrange as $y^2 = 4(x^2 - 1)$, or equivalent	A1	[6]

## Question 5:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Separate variables correctly | B1 | |
| Integrate and obtain term $\ln x$ | B1 | |
| Integrate and obtain term $\frac{1}{2}\ln(y^2 + 4)$ | B1 | |
| Evaluate a constant or use limits $y = 0$, $x = 1$ in a solution containing $a\ln x$ and $b\ln(y^2+4)$ | M1 | |
| Obtain correct solution in any form, e.g. $\frac{1}{2}\ln(y^2+4) = \ln x + \frac{1}{2}\ln 4$ | A1 | |
| Rearrange as $y^2 = 4(x^2 - 1)$, or equivalent | A1 | [6] |

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5 Given that $y = 0$ when $x = 1$, solve the differential equation

$$x y \frac { \mathrm {~d} y } { \mathrm {~d} x } = y ^ { 2 } + 4 ,$$

obtaining an expression for $y ^ { 2 }$ in terms of $x$.

\hfill \mbox{\textit{CAIE P3 2010 Q5 [6]}}

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Q1 4 Q2 6 Q3 6 Q4 6 Q5 6 Q6 8 Q7 9 Q8 9 Q9 9 Q10 12