Lecture 3: Exercise 4
Explanation
This exercise provides some more practice in proving that a proposed solution to a differential equation is indeed a solution. This also
establishes the general solution of the harmonic oscillator problem.
Hint
You will need the sum rule and the trigonoemtric rules of differentiation.
Answer
We begin by writing,
xHtL = A cos Ω t + B sin Ω t.
We can find the velocity by differentiating,
d
vHtL =
dt
or,
d
xHtL =
d
A cos Ω t +
dt
B sin Ω t
dt
2
l3e4.cdf
d
d
vHtL = A
cos Ω t + B
dt
sin Ω t
dt
or,
vHtL = -A Ω sin Ω t + B Ω cos Ω t.
We can find the acceleration,
d
aHtL =
d
d
vHtL = dt
A Ω sin Ω t +
dt
B Ω cos Ω t
dt
or,
d
aHtL = -A Ω
d
sin Ω t + B Ω
dt
cos Ω t
dt
or
aHtL = -A Ω2 cos Ω t - B Ω2 sin Ω t
we can factor the Ω2 ,
aHtL = -HA cos Ω t - B sin Ω tL Ω2 = -Ω2 xHtL,
which is the form of Eq. (6). When t = 0,
xH0L = A cos 0 + B sin 0 = A.
and
vH0L = -A Ω sin 0 + B Ω cos 0 = B Ω.
If we label the initial position v0 and the initial velocity v0 , then
x0 = A
and
v0 = B Ω.