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7.1. AN INTRODUCTION TO DIFFERENTIAL EQUATIONS
Differential equations in the world around us
As we have noted, differential equations give a natural way to describe phenomena we see
in the real world. For instance, physical principles are frequently expressed as a description
of how a quantity changes. A good example is Newton’s Second Law, an important physcial
principle that says:
The product of an object’s mass and acceleration equals the force applied to it.
For instance, when gravity acts on an object near the earth’s surface, it exerts a force
equal to mg, the mass of the object times the gravitational constant g. We therefore have
ma = mg, or
dv
dt
= g,
where v is the velocity of the object, and g = 9.8 meters per second squared. Notice that
this physical principle does not tell us what the object’s velocity is, but rather how the
object’s velocity changes.
Activity 7.2.
Shown below are two graphs depicting the velocity of falling objects. On the left is
the velocity of a skydiver, while on the right is the velocity of a meteorite entering the
Earth’s atmosphere.
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t
v
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v
Skydiver’s velocity
Meteorite’s velocity