4—Differential Equations 87The two second order differential equations have four arbitrary constants in their solution. You
can specify the initial values of two positions and of two velocities this way. As a specific example
suppose that all initial velocities are zero and that the first mass is pushed to coordinatex 0 and
released.
x 1 (0) =x 0 =A 1 +A 2 +A 3 +A 4
x 2 (0) = 0 =A 1 +A 2 −A 3 −A 4
vx 1 (0) = 0 =iω 1 A 1 −iω 1 A 2 +iω 2 A 3 −iω 2 A 4
vx 2 (0) = 0 =iω 1 A 1 −iω 1 A 2 −iω 2 A 3 +iω 2 A 4 (4.53)
With a little thought (i.e. don’t plunge blindly ahead) you can solve these easily.
A 1 =A 2 =A 3 =A 4 =
x 0
4
x 1 (t) =
x 0
4
[
eiω^1 t+e−iω^1 t+eiω^2 t+e−iω^2 t
]
=
x 0
2
[
cosω 1 t+ cosω 2 t
]
x 2 (t) =
x 0
4
[
eiω^1 t+e−iω^1 t−eiω^2 t−e−iω^2 t
]
=
x 0
2
[
cosω 1 t−cosω 2 t
]
From the results of problem3.34, you can rewrite these as
x 1 (t) =x 0 cos
(
ω 2 +ω 1
2
t
)
cos(
ω 2 −ω 1
2
t
)
x 2 (t) =x 0 sin
(
ω 2 +ω 1
2
t
)
sin(
ω 2 −ω 1
2
t
)
(4.54)
As usual you have to draw some graphs to understand what these imply. If the center springk 3
is a lot weaker than the outer ones, then Eq. (4.51) implies that the two frequencies are close to each
other and so|ω 1 −ω 2 | ω 1 +ω 2. Examine Eq. (4.54) and you see that one of the two oscillating
factors oscillate at a much higher frequency than the other. To sketch the graph ofx 2 for example you
should draw one factor
[
sin(
(ω 2 +ω 1 )t/ 2
)]
and the other factor[
sin(
(ω 2 −ω 1 )t/ 2
)]
and graphically
multiply them.
x 2
The massm 2 starts without motion and its oscillations gradually build up. Later they die down
and build up again (though with reversed phase). Look at the other mass, governed by the equation
forx 1 (t)and you see that the low frequency oscillation from the(ω 2 −ω 1 )/ 2 part is big where the one
forx 2 is small and vice versa. The oscillation energy moves back and forth from one mass to the other.
4.11 Legendre’s Equation
This equation and its solutions appear when you solve electric and gravitational potential problems
in spherical coordinates [problem9.20]. They appear when you study Gauss’s method of numerical
integration [Eq. (11.27)] and they appear when you analyze orthogonal functions [problem6.7]. Because
it shows up so often it is worth the time to go through the details in solving it.
[