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Date May Example question Marks available 1 Reference code EXM.3.AHL.TZ0.2
Level Additional Higher Level Paper Paper 3 Time zone Time zone 0
Command term State Question number 2 Adapted from N/A

Question

This question will investigate the solution to a coupled system of differential equations when there is only one eigenvalue.

It is desired to solve the coupled system of differential equations

˙x=3x+y

˙y=x+y.

The general solution to the coupled system of differential equations is hence given by

(xy)=A(11)e2t+B(tt+1)e2t

As t the trajectory approaches an asymptote.

Show that the matrix (3111) has (sadly) only one eigenvalue.  Find this eigenvalue and an associated eigenvector.

[7]
a.

Hence, verify that (xy)=(11)e2t is a solution to the above system.

[5]
b.

Verify that (xy)=(tt+1)e2t is also a solution.

[5]
c.

If initially at t=0,x=20,y=10 find the particular solution.

[3]
d.

Find the values of x and y when t=2.

[2]
e.

 Find the equation of this asymptote.

[3]
f.i.

State the direction of the trajectory, including the quadrant it is in as it approaches this asymptote.

[1]
f.ii.

Markscheme

|3λ111λ|=0(3λ)(1λ)+1=0      M1A1

λ24λ+4=0(λ2)2=0      A1A1

So only one solution    λ=2      AGA1

(1111)(pq)=(00)p+q=0      M1

So an eigenvector is (11)      A1

[7 marks]

a.

(3111)(11)=2(11)

So (3x+yx+y)=(3111)(xy)=(3111)(11)e2t=2(11)e2t            M1A1A1

and (xy)=(11)e2t(˙x˙y)=(11)2e2t      M1A1

showing that (xy)=(11)e2t is a solution      AG

[5 marks]

b.

(3x+yx+y)=(3tt+1tt+1)e2t=(2t+12t+1)e2t          M1A1

(˙x˙y)=(e2t+t2e2te2t+(t+1)2e2t)=(2t+12t+1)e2t      M1A1A1

Verifying that (xy)=(tt+1)e2t is also a solution      AG

[5 marks]

c.

Require (2010)=A(11)+B(01)A=20,B=30           M1A1

(xy)=20(11)e2t+30(tt+1)e2t    A1

[3 marks]

d.

t=2x=4370,y=2730(3sf)      A1A1

[2 marks]

e.

As t,x30te2t,y30te2t      M1A1

so asymptote is y=x      A1

[3 marks]

f.i.

Will approach the asymptote in the 4th quadrant, moving away from the origin.      R1

[1 mark]

f.ii.

Examiners report

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f.i.
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f.ii.

Syllabus sections

Topic 5—Calculus » AHL 5.17—Phase portrait
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