We only get non-trivial values for (x 0 ,y 0 ) if the determinant of coefficients is zero:
∣
∣
∣
∣
−λ 1
−ω^2 −λ
∣
∣
∣
∣=
∣
∣
∣
∣
[
01
−ω^20
]
−λ
[
10
01
]∣∣
∣
∣=^0
This is precisely the eigenvalue equation for thematrix of coefficientsof the original
system:
A=
[
01
−ω^20
]
Convert the equation
d^2 x
dt^2
+ 4 x= 0
to a first order system as described above.
Try a solution of the formx=x 0 eλt,y=y 0 eλtin this system and find the possible
values ofλthat will yield a non-trivial solution – they will in fact be complex. Hence
solve the equations and express the solutions without complex numbers.
13.10 Answers to reinforcement exercises
- (i)
[
2 − 1
12
][
x
y
]
=
[
− 1
0
]
(ii)
[ 3 − 12
223
− 30 − 1
][x
y
z
]
=
[ 1
2
− 3
]
(iii)
[
123
1 − 1 − 1
][a
b
c
]
=
[
1
3
]
(iv)
[ 4 − 2
33
1 − 1
][
u
v
]
=
[ 1
2
− 1
]
2.IfA=
[
− 123
402
− 112
]
B=
[
013
2 − 14
3 − 12
]
(i) 2,−1, 2, 0, 2,− 1
(ii) 11,− 4
(iii) (a)
[ − 21 − 3
23 − 8
− 11 5 − 2
]
(b)
[ 617
− 61016
303
]
(c)
[ 13 − 611
6216
8 − 45
]
(d)
[ 13 8
− 10 8 12
−98 11
]
