Lecture Notes in Differential Equations - Bruce E. Shapiro

Lesson 2
A Geometric View
One way to look at a differential equation is as a description of a trajectory
or position of an object over time. We will steal the term “particle” from
physics for this idea. By a particle we will mean a “thing” or “object” (but
doesn’t sound quite so coarse) whose location at time t = t 0 is given by
y = y 0 (2.1)
At a later time t > t 0 we will describe the position by a function
y = φ(t) (2.2)
which we will generally write as y(t) to avoid the confusion caused by the
extra Greek symbol. 1 We can illustrate this in the following example.
Example 2.1. Find y(t) for all t > 0 if dy/dt = y and y(0) = 1.
In example 1.5 we found that the general solution of the differential equation
is
y = Ce t (2.3)
We can determine the value of C from the initial condition, which tells us
that y = 1 when t = 1:
Hence the solution of the initial value problem is
1 = y(0) = Ce 0 = C (2.4)
y = e t (2.5)
1 Mathematically, we mean that φ(t) is a solution of the equations that describes what
happens to y as a result of some differential equation dy/dt = f(t, y); in practice, the
equation for φ(t) is identical to the equation for y(t) and the distinction can be ignored.
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