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(1−1)e2t+B(t−t+1)e2t
As t→∞ the trajectory approaches an asymptote.
Show that the matrix (31−11) has (sadly) only one eigenvalue. Find this eigenvalue and an associated eigenvector.
Hence, verify that (xy)=(1−1)e2t is a solution to the above system.
Verify that (xy)=(t−t+1)e2t is also a solution.
If initially at t=0,x=20,y=10 find the particular solution.
Find the values of x and y when t=2.
Find the equation of this asymptote.
State the direction of the trajectory, including the quadrant it is in as it approaches this asymptote.
Markscheme
|3−λ1−11−λ|=0⇒(3−λ)(1−λ)+1=0 M1A1
λ2−4λ+4=0⇒(λ−2)2=0 A1A1
So only one solution λ=2 AGA1
(11−1−1)(pq)=(00)⇒p+q=0 M1
So an eigenvector is (1−1) A1
[7 marks]
(31−11)(1−1)=2(1−1)
So (3x+y−x+y)=(31−11)(xy)=(31−11)(1−1)e2t=2(1−1)e2t M1A1A1
and (xy)=(1−1)e2t⇒(˙x˙y)=(1−1)2e2t M1A1
showing that (xy)=(1−1)e2t is a solution AG
[5 marks]
(3x+y−x+y)=(3t−t+1−t−t+1)e2t=(2t+1−2t+1)e2t M1A1
(˙x˙y)=(e2t+t2e2t−e2t+(−t+1)2e2t)=(2t+1−2t+1)e2t M1A1A1
Verifying that (xy)=(t−t+1)e2t is also a solution AG
[5 marks]
Require (2010)=A(1−1)+B(01)⇒A=20,B=30 M1A1
(xy)=20(1−1)e2t+30(t−t+1)e2t A1
[3 marks]
t=2⇒x=4370,y=−2730(3sf) A1A1
[2 marks]
As t→∞,x≃30te2t,y≃−30te2t M1A1
so asymptote is y=−x A1
[3 marks]
Will approach the asymptote in the 4th quadrant, moving away from the origin. R1
[1 mark]