A2 June 2025 Paper 2 Q6

AQACurrent spec3 marksFirst Order Differentials

6 A curve passes through the point \((2, k)\), where \(k \gt 1\)

The curve satisfies the differential equation

\[\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{x^2 - 3y}{xy}\]

Using Euler’s step by step method once, with starting point \((2, k)\) and a step length of 0.1, gives an estimate of \(y = 6.069\) when \(x = 2.1\)

Find the value of \(k\)

Give your answer to three decimal places. [3 marks]

Mark scheme

Scheme	Marks	AO
Substitutes into Euler’s formula with 6.069 as the subject (allow one slip).	M1	1.1a
Using Euler’s formula, forms and obtains a solution to a quadratic equation in \(k\)	M1	1.1a
Deduces AWRT 6.187 Accept no mention made of rejected root, or root rejected for the wrong reason.	A1	2.2a
	(3 marks)

Typical solution

\[y_1 = y_0 + h\mathrm{f}(x_0)\]\[6.069 = k + \frac{0.1(4 - 3k)}{2k}\]\[6.069 = k - 0.15 + \frac{0.2}{k}\]\[k^2 - 6.219k + 0.2 = 0\]\[k = 6.187 \text{ or } k = 0.032\]\[k \gt 1 \Rightarrow k = 6.187\]