Second Order Linear Differential Equations

Example 1. Find an equation for the oscillatory motion of an object, given that the period
is 2π/3 and at time t =0,y=1,y ′ =3.
SOLUTION In general the period is 2π/ω, so that here
2π
ω
= 2π
3
The equation of motion takes the form
and therefore ω =3.
y(t) =A sin (3ω + φ0).
Differentiating the equation of motion gives
Applying the initial conditions, we have
and therefore
y ′ (t) =3A cos(3t + φ0).
y(0) = 1 = A sin φ0, y ′ (0) = 3 = 3A cos φ0
A sin φ0 =1, Acos φ0 =1.
Adding the squares of these equations, we have
Since A>0, A= √ 2.
2=A 2 sin 2 φ0 + A 2 cos 2 φ0 = A 2 .
Finally, to find φ0, note that
√
2 sin φ0 = 1 and √ 2 cos φ0 =1.
These equations imply that φ0 = π/4. Thus, the equation of motion is
Damped Vibrations
y(t) = √ 2 sin (3t + 1 4
π).
If the spring is not frictionless or if there the surrounding medium resists the motion of the
object (for example, air resistance), then the resistance tends to dampen the oscillations.
Experiments show that such a resistant force R is approximately proportional to the
velocity v = y ′ and acts in a direction opposite to the motion:
R = −cy ′ with c>0.
Taking this force into account, the force equation reads
F = −ky(t) − cy ′ (t).
108