Chapter 1
Geometric Approach to
Differential Equations
In a basic elementary differential equations course, the emphasis is on linear dif-
ferential equations. One example treated is the linear second-order differential
equation
(1.1) m ¨ x + b ˙ x + k x = 0,
where we write ˙ x for
dx
dt
and ¨ x for
d2x
dt2
, and m, k 0 and b ≥ 0. This equation
is a model for a linear spring with friction and is also called a damped harmonic
oscillator.
To determine the future motion, we need to know both the current position
x and the current velocity ˙. x Since these two quantities determine the motion,
it is natural to use them for the coordinates; writing v = ˙, x we can rewrite this
equation as a linear system of first-order differential equations (involving only first
derivatives), so m ˙ v = m ¨ x = −k x − b v:
˙ x = v, (1.2)
˙ v = −
k
m
x −
b
m
v.
In matrix notation, this becomes
˙ x
˙ v
=
0 1
−
k
m
−
b
m
x
v
.
For a linear differential equation such as (1.1), the usual solution method is to
seek solutions of the form x(t) = eλ t, for which ˙ x = λ eλ t and ¨ x = λ2 eλ t. We
need
m
λ2 eλ t
+ b λ
eλ t
+ k
eλ t
= 0,
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