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Date November 2011 Marks available 15 Reference code 11N.3ca.hl.TZ0.6
Level HL only Paper Paper 3 Calculus Time zone TZ0
Command term Find Question number 6 Adapted from N/A

Question

The real and imaginary parts of a complex number x+iy are related by the differential equation (x+y)dydx+(xy)=0.

By solving the differential equation, given that y=3 when x =1, show that the relationship between the modulus r and the argument θ of the complex number is r=2eπ3θ.

Markscheme

(x+y)dydx+(xy)=0

dydx=yxx+y

let y=vx     M1

dydx=v+xdvdx     A1

v+xdvdx=vxxx+vx     (A1)

xdvdx=v1v+1v=v1v2vv+1=1v21+v     A1

v+11+v2dv=1xdx     M1

v1+v2dv+11+v2dv=1xdx     M1

12ln|1+v2|+arctanv=ln|x|+k     A1A1

Notes: Award A1 for 12ln|1+v2|, A1 for the other two terms.

Do not penalize missing k or missing modulus signs at this stage.

 

12ln|1+y2x2|+arctanyx=ln|x|+k     M1

12ln4+arctan3=ln1+k     (M1)

k=ln2+π3     A1

12ln|1+y2x2|+arctanyx=ln|x|+ln2+π3

attempt to combine logarithms     M1

12ln|y2+x2x2|+12ln|x2|=ln2+π3arctanyx

12ln|y2+x2|=ln2+π3arctanyx     (A1)

y2+x2=eln2+π3arctanyx     (A1)

y2+x2=eln2×eπ3arctanyx     A1

r=2eπ3θ     AG

[15 marks]

Examiners report

Most candidates realised that this was a homogeneous differential equation and that the substitution y=vx was the way forward. Many of these candidates reached as far as separating the variables correctly but integrating v+1v2+1 proved to be too difficult for many candidates – most failed to realise that the expression had to be split into two separate integrals. Some candidates successfully evaluated the arbitrary constant but the combination of logs and the subsequent algebra necessary to obtain the final result proved to be beyond the majority of candidates.

Syllabus sections

Topic 9 - Option: Calculus » 9.5 » First-order differential equations.
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