Question 1:
Part a:
𝑥̈ (𝑡) + 𝜔𝑛2 𝑥(𝑡) = 0
(1)
𝑥(𝑡) = 𝑒 𝑟𝑡 , 𝑥̇ (𝑡) = 𝑟𝑒 𝑟𝑡 , 𝑥̈ (𝑡) = 𝑟 2 𝑒 𝑟𝑡
(2)
𝑟 2 𝑒 𝑟𝑡 + 𝜔𝑛2 𝑒 𝑟𝑡 = 0
(3)
𝑒 𝑟𝑡 (𝑟 2 + 𝜔2 ) = 0
(4)
𝑟 2 + 𝜔𝑛2 = 0
(5)
𝑟 = ±𝜔𝑛 𝑖
(6)
𝑥(𝑡) = 𝐴𝑒 𝜔𝑛𝑖𝑡 + 𝐵𝑒 −𝜔𝑛𝑖𝑡
(7)
Plugging (2) into (1):
Finding the roots:
Using Euler’s Formula and plugging into (7):
𝑥(𝑡) = 𝐴(cos(𝜔𝑛 𝑡) + 𝑖 sin(𝜔𝑛 𝑡)) + 𝐵(cos(−𝜔𝑛 𝑡) + 𝑖 sin(−𝜔𝑛 𝑡))
(8)
𝑥(𝑡) = (𝐴 + 𝐵) cos(𝜔𝑛 𝑡) + 𝑖(𝐴 − 𝐵) sin(𝜔𝑛 𝑡)
(9)
𝑥(𝑡) = 𝐶 cos(𝜔𝑛 𝑡) + 𝐷 sin(𝜔𝑛 𝑡)
(10)
Using initial conditions of (𝑥(0), 𝑥̇ (0)) = (𝑥0 , 𝑣0 ), 𝐶 ad 𝐷 can be found:
𝑥(0) = 𝑥0 = 𝐶
(11)
𝑥̇ (𝑡) = −𝜔𝑛 𝐶 sin(𝜔𝑛 𝑡) + 𝜔𝑛 𝐷 cos(𝜔𝑛 𝑡)
(12)
𝑥̇ (0) = 𝑣𝑜 = 𝜔𝑛 𝐷
(13)
𝐷=
𝑣0
𝜔𝑛
(14)
Plugging (11) and (14) back into (10):
𝑥(𝑡) = 𝑥0 cos(𝜔𝑛 𝑡) +
𝑣0
sin(𝜔𝑛 𝑡)
𝜔𝑛
(15)
Part b:
𝑥1
𝑥(𝑡)
[𝑥 ] = [
]
𝑥̇ (𝑡)
2
(16)
𝑥2
𝑥̇
𝑥̇ (𝑡)
[ 1] = [
]=[
]
−𝜔𝑛2 𝑥1
𝑥̇ 2
𝑥̈ (𝑡)
(17)
Two first order equations using 𝑥(𝑡) and 𝑥̇ (𝑡) as state space variables are:
𝑥2
𝑥̇
[ 1] = [ 2 ]
−𝜔𝑛 𝑥1
𝑥̇ 2
Part c:
See MATLAB code.
Part d:
See MATLAB code.
(18)