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.
Given the initial conditions that when t=0, x=150, y=50, find the particular solution.
Hence find the solution when t=1.
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.
Markscheme
The characteristic equation is given by
|3−λ223−λ|=0⇒λ2−6λ+5=0⇒λ=1 or 5 M1A1A1A1
λ=1 (2222)(pq)=(00) gives an eigenvector of form (1−1) M1A1
λ=5 (−222−2)(pq)=(00) gives an eigenvector of form (11) M1A1
General solution is (xy)=Aet(1−1)+Be5t(11) A1A1
[10 marks]
Require A+B=150, −A+B=50⇒A=50, B = 100 M1A1
Particular solution is (xy)=50et(1−1)+100e5t(11) A1
[3 marks]
t=1⇒(xy)=(1500014700) (3sf) A1
[1 mark]
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]