A2 June 2024 Paper 2 Q9

AQACurrent spec4 marksFirst Order Differentials

9 A curve passes through the point \((-2, 4.73)\) and satisfies the differential equation

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

Use Euler’s step by step method once, and then the midpoint formula

\[y_{r+1} = y_{r-1} + 2h\mathrm{f}(x_r, y_r), \quad x_{r+1} = x_r + h\]

once, each with a step length of 0.02, to estimate the value of \(y\) when \(x = -1.96\)

Give your answer to five significant figures. [4 marks]

Mark scheme

Scheme	Marks	AO
Uses Euler’s method once to calculate an estimate of \(y\) when \(x = -1.98\) Condone one error.	M1	1.1a
Obtains AWRT 4.766 PI by final AWRT 4.8027	A1	1.1b
Uses midpoint formula once with their \(y_1\) Condone one error.	M1	3.1a
Obtains AWRT 4.8027	A1	1.1b
	(4 marks)

Typical solution

\[x_0 = -2\]\[y_0 = 4.73\]\[h = 0.02\]\[y_1 = 4.73 + 0.02\left(\frac{4.73^2 - (-2)^2}{2(-2) + 3(4.73)}\right)\]\[= 4.766060648\]\[x_1 = -1.98\]\[y_2 = 4.73 + 0.04\left(\frac{4.766060648^2 - (-1.98)^2}{2(-1.98) + 3(4.766060648)}\right)\]\[= 4.8027\]