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Date November 2018 Marks available 6 Reference code 18N.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=1+yx, where x0.

Consider the family of curves which satisfy the differential equation dydx=1+yx, where x0.

Given that y(1)=1, use Euler’s method with step length h = 0.25 to find an approximation for y(2). Give your answer to two significant figures.

[4]
a.

Solve the equation dydx=1+yx for y(1)=1.

[6]
b.

Find the percentage error when y(2) is approximated by the final rounded value found in part (a). Give your answer to two significant figures.

[3]
c.

Find the equation of the isocline corresponding to dydx=k, where k0kR.

[1]
d.i.

Show that such an isocline can never be a normal to any of the family of curves that satisfy the differential equation.

[4]
d.ii.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

attempt to apply Euler’s method         (M1)

xn+1=xn+0.25;yn+1=yn+0.25×(1+ynxn)

     (A1)(A1)

Note: Award A1 for correct x values, A1 for first three correct y values.

 

y = 3.3      A1

 

[4 marks]

a.

METHOD 1

I(x)=e1xdx       (M1)

=elnx

=1x       (A1)

1xdydxyx2=1x       (M1)

ddx(yx)=1x

yx=ln|x|+C       A1

y(1)=1C=1       M1

y=xln|x|+x       A1

 

METHOD 2

v=yx       M1

dvdx=1xdydx1x2y       (A1)

v+xdvdx=1+v       M1

1dv=1xdx

v=ln|x|+C

yx=ln|x|+C       A1

y(1)=1C=1       M1

y=xln|x|+x       A1

 

[6 marks]

b.

y(2)=2ln2+2=3.38629

percentage error =3.386293.33.38629×100      (M1)(A1)

= 2.5%       A1

 

[3 marks]

c.

dydx=k1+yx=k      A1

y=(k1)x

 

[1 mark]

d.i.

gradient of isocline equals gradient of normal        (M1)

k1=1k or k(k1)=1      A1

k2k+1=0       A1

Δ=14<0       R1

no solution       AG

Note: Accept alternative reasons for no solutions.

 

[4 marks]

d.ii.

Examiners report

[N/A]
a.
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b.
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c.
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d.i.
[N/A]
d.ii.

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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