Sketch, on one diagram, the four isoclines corresponding to $f(x,{\text{ }}y) = k$ where $k$ takes the values $-1$, $-0.5$, $0$ and $1$. Indicate clearly where each isocline crosses the $y$ axis.

A curve, $C$, passes through the point $(0,1)$ and satisfies the differential equation above.

Sketch $C$ on your diagram.

A curve, $C$, passes through the point $(0,1)$ and satisfies the differential equation above.

State a particular relationship between the isocline $f(x,{\text{ }}y) =  - 0.5$ and the curve $C$, at their point of intersection.

A curve, $C$, passes through the point $(0,1)$ and satisfies the differential equation above.

Use Euler’s method with a step interval of $0.1$ to find an approximate value for $y$ on $C$, when $x = 0{\text{.}}5$.

A1 for 4 parallel straight lines with a positive gradient     A1

A1 for correct $y$ intercepts     A1

[2 marks]

A1 for passing through $(0,1)$ with positive gradient less than $2$

A1 for stationary point on $y = 2x$

A1 for negative gradient on both of the other $2$ isoclines     A1A1A1

[3 marks]

The isocline is perpendicular to $C$     R1

[1 mark]

${y_{n + 1}} = {y_n} + 0.1({y_n} - 2{x_n})\;\;\;( = 1.1{y_n} - 0.2{x_n})$     (M1)(A1)

Note:     Also award M1A1 if no formula seen but ${y_2}$ is correct.

${y_0} = 1,{\text{ }}{y_1} = 1.1,{\text{ }}{y_2} = 1.19,{\text{ }}{y_3} = 1.269,{\text{ }}{y_4} = 1.3359$     (M1)

${y_5} = 1.39{\text{ to 3sf}}$     A1

Note:     M1 is for repeated use of their formula, with steps of $0.1$.

Note:     Accept $1.39$ or $1.4$ only.

[4 marks]

Total [10 marks]

Some candidates ignored the instruction to prove from first principles and instead used standard differentiation. Some candidates also only found a derivative from one side.

Parts (b) and (c) were attempted by very few candidates. Few recognized that the gradient of the curve had to equal the value of $k$ on the isocline.

Parts (b) and (c) were attempted by very few candidates. Few recognized that the gradient of the curve had to equal the value of $k$ on the isocline.

Those candidates who knew the method managed to score well on this part. On most calculators a short program can be written in the exam to make Euler’s method very quick. Quite a few candidates were losing time by calculating and writing out many intermediate values, rather than just the $x$ and$y$ values.