Loading [MathJax]/jax/element/mml/optable/BasicLatin.js

User interface language: English | Español

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

Question

Consider the system of paired differential equations

˙x=3x+2y

˙y=2x+3y.

This represents the populations of two species of symbiotic toadstools in a large wood.

Time t is measured in decades.

Use the eigenvalue method to find the general solution to this system of equations.

[10]
a.

Given the initial conditions that when t=0x=150y=50, find the particular solution.

[3]
b.i.

Hence find the solution when t=1.

[1]
b.ii.

As t, find an asymptote to the trajectory of the particular solution found in (b)(i) and state if this trajectory will be moving towards or away from the origin.

[4]
c.

Markscheme

The characteristic equation is given by

|3λ223λ|=0λ26λ+5=0λ=1 or 5     M1A1A1A1

λ=1 (2222)(pq)=(00) gives an eigenvector of form (11)     M1A1

λ=5 (2222)(pq)=(00) gives an eigenvector of form (11)     M1A1

General solution is (xy)=Aet(11)+Be5t(11)      A1A1

[10 marks]

a.

Require A+B=150, A+B=50A=50, B = 100       M1A1

Particular solution is (xy)=50et(11)+100e5t(11)       A1

[3 marks]

b.i.

t=1(xy)=(1500014700) (3sf)      A1

[1 mark]

b.ii.

The dominant term is 100e5t(11)     so as t(xy)100e5t(11)     M1A1

Giving the asymptote as y=x     A1

The trajectory is moving away from the origin.       A1

[4 marks]

c.

Examiners report

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

Syllabus sections

Topic 5—Calculus » AHL 5.17—Phase portrait
Show 63 related questions
Topic 5—Calculus

View options