Chapter 1

Differential Equations

and Their Solutions

1.1. First-Order ODE: Existence and Uniqueness

What does the following sentence mean, and what image should it

cause you to form in your mind?

Let V :

Rn

× R →

Rn

be a time-dependent vector

field, and let x(t) be a solution of the differential

equation

dx

dt

= V (x, t) satisfying the initial condition

x(t0) = x0.

Let us consider a seemingly very different question. Suppose you

know the wind velocity at every point of space and at all instants of

time. A puff of smoke drifts by, and at a certain moment you note the

precise location of a particular smoke particle. Can you then predict

where that particle will be at all future times?

We will see that when this somewhat vague question is trans-

lated appropriately into precise mathematical concepts, it leads to

the above “differential equation”, and that the answer to our predic-

tion question translates to the central existence and uniqueness result

in the theory of differential equations. (The answer, by the way, turns

out to be a qualified “yes”, with several important caveats.)

We interpret “space” to mean the n-dimensional real number

space

Rn,

so a “point of space” is just an n-tuple x = (x1,... , xn) of

real numbers. If you feel more comfortable thinking n = 3, that’s fine

5

http://dx.doi.org/10.1090/stml/051/02