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Date November 2019 Marks available 10 Reference code 19N.3.AHL.TZ0.Hca_4
Level Additional Higher Level Paper Paper 3 Time zone Time zone 0
Command term Solve Question number Hca_4 Adapted from N/A

Question

Consider the differential equation dydx=4x2+y2xyx2, with y=2 when x=1.

Use Euler’s method, with step length h=0.1, to find an approximate value of y when x=1.4.

[5]
a.

Sketch the isoclines for dydx=4.

[3]
b.

Express m22m+4 in the form (ma)2+b , where abZ.

[1]
c.i.

Solve the differential equation, for x>0, giving your answer in the form y=f(x).

[10]
c.ii.

Sketch the graph of y=f(x) for 1x1.4 .

[1]
c.iii.

With reference to the curvature of your sketch in part (c)(iii), and without further calculation, explain whether you conjecture f(1.4) will be less than, equal to, or greater than your answer in part (a).

[2]
c.iv.

Markscheme

       (M1)(A1)(A1)(A1)A1

y(1.4)5.34

Note: Award A1 for each correct y value.
For the intermediate y values, accept answers that are accurate to 2 significant figures.
The final y value must be accurate to 3 significant figures or better.

[5 marks]

a.

attempt to solve 4x2+y2xyx2=4        (M1)

y2xy=0

y(yx)=0

y=0  or  y=x

        A1A1

[3 marks]

b.

m22m+4=(m1)2+3(a=1,b=3)        A1

[1 mark]

c.i.

recognition of homogeneous equation,
let y=vx             M1

the equation can be written as

v+xdvdx=4+v2v         (A1)

xdvdx=v22v+4

1v22v+4dv=1xdx             M1

Note: Award M1 for attempt to separate the variables.

1(v1)2+3dv=1xdx from part (c)(i)             M1

13arctan(v13)=lnx(+c)          A1A1

x=1,y=2v=2

13arctan(13)=ln1+c             M1

Note: Award M1 for using initial conditions to find c.

c=π63(=0.302)        A1

arctan(v13)=3lnx+π6

substituting v=yx             M1

Note: This M1 may be awarded earlier.

y=x(3tan(3lnx+π6)+1)        A1

[10 marks]

c.ii.

curve drawn over correct domain       A1

 

[1 mark]

c.iii.

the sketch shows that f is concave up       A1

Note: Accept f is increasing.

this means the tangent drawn using Euler’s method will give an underestimate of the real value, so f(1.4) > estimate in part (a)       R1

Note: The R1 is dependent on the A1.

[2 marks]

c.iv.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.i.
[N/A]
c.ii.
[N/A]
c.iii.
[N/A]
c.iv.

Syllabus sections

Topic 5 —Calculus » AHL 5.18—1st order DE’s – Euler method, variables separable, integrating factor, homogeneous DE using sub y=vx
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Topic 5 —Calculus

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