36600 MA LECTURE NOTES: FRIDAY, APRIL 3
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Then there exists a unique solution on the interval I . This is known as the Existence and Uniqueness Theorem
for Linear Systems. The statement for linear systems follows from that for nonlinear systems because if we
denote
n
Gi
Gi (t, x1 , x2 , . . . , xn ) =
pij (t) xj + gi (t)
=
= pij (t).
xj
j =1
Note the similarity here with the Existence and Uniqueness Theorem for second order linear equations
presented in Lecture #14 on Wednesday, October 1.
Applications
Spring-Mass System: One Mass, One Spring. We discuss several examples of oscillations with springs,
each increasing in complexity. For simplicity, we assume that
(a) the original length of the spring is negligible to the motion of the mass, so that Hookes Law states
that the restoring force is Fs = k u in terms a of displacement u;
(b) all motion takes place on a frictionless track so that we can ignore any frictional forces; and
(c) that the motion is not aected of the force of gravity.
First, say that we have a mass m attached to a spring with constant k . Assume that the mass is under
the inuence of an external force F (t). Denote u = u(t) as the position of the mass at time t. Equilibrium
is seen to be at u = 0, so this also denotes the displacement from equilibrium of the mass at time t. Hookes
Law states that the restoring force of the spring on the mass is
Fs = k u.
Newtons Second Law of Motion states that F = m a is the sum of the forces on the mass:
d2 u
m 2 = k u + F (t).
dt
This can be expressed as a system of rst order dierential equations. Indeed, if we denote
x1 = u
x = x2
1
=
x2 = u
m x = k x1 + F (t)
2