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Date May 2016 Marks available 4 Reference code 16M.3ca.hl.TZ0.4
Level HL only Paper Paper 3 Calculus Time zone TZ0
Command term Show that Question number 4 Adapted from N/A

Question

Consider the differential equation dydx=xyxy where y>0 and y=2 when x=0.

Show that putting z=y2 transforms the differential equation into dzdx+2xz=2x.

[4]
a.

By solving this differential equation in z, obtain an expression for y in terms of x.

[9]
b.

Markscheme

METHOD 1

z=y2y=z1/2

dydx=12z1/2dzdx    M1A1

substituting, 12z1/2dzdx=xz1/2xz1/2     M1A1

dzdx+2xz=2x    AG

METHOD 2

z=y2

dzdx=2ydydx    M1A1

dzdx=2x2xy2    M1A1

dzdx=2xz=2x    AG

[4 marks]

a.

METHOD 1

integrating factor =e2xdx=ex2     (M1)A1

ex2dzdx+2xex2z=2xex2    (M1)

zex2=2xex2dx    A1

=ex2+C    A1

substitute y=2 therefore z=4 when x=0     (M1)

4=1+C

C=3    (A1)

the solution is z=1+3ex2     (M1)

 

Note:     This line may be seen before determining the value of C.

 

so that y=1+3ex2     A1

METHOD 2

dzdx=2x(1z)

11zdz=2xdx    M1

ln(1z)=x2+C    A1A1

1z=ex2c (or 1z=Bex2)     M1A1

solving for z     (M1)

z=1+Aex2

z=4 when x=0     (M1)

so A=3     (A1)

the solution is z=1+3ex2

so y=1+3ex2     A1

[9 marks]

b.

Examiners report

Several misconceptions were identified that showed poor understanding of the chain rule. Although many candidates were successful in establishing the result the presentation of their work was far from what is expected in a show that question.

a.

Part (b) was well attempted using both method 1 (integration factor) and 2 (separation of variables). The most common error was omission of the constant of integration or errors in finding its value. Candidates that used method 2 often had difficulties in integrating 1(1z) correctly and making z the subject often losing out on accuracy marks.

b.

Syllabus sections

Topic 9 - Option: Calculus
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