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Date None Specimen Marks available 7 Reference code SPNone.3ca.hl.TZ0.2
Level HL only Paper Paper 3 Calculus Time zone TZ0
Command term Find Question number 2 Adapted from N/A

Question

Consider the differential equation

xdydx=y+x2y2, x>0, x2>y2.

Show that this is a homogeneous differential equation.

[1]
a.

Find the general solution, giving your answer in the form y=f(x) .

[7]
b.

Markscheme

the equation can be rewritten as

dydx=y+x2y2x=yx+1(yx)2     A1

so the differential equation is homogeneous     AG

[1 mark]

a.

put y = vx so that dydx=v+xdvdx     M1A1

substituting,

v+xdvdx=v+1v2     M1

dv1v2=dxx     M1

arcsinv=lnx+C     A1

yx=sin(lnx+C)     A1

y=xsin(lnx+C)     A1

[7 marks]

b.

Examiners report

[N/A]
a.
[N/A]
b.

Syllabus sections

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