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 :
× R
be a time-dependent vector
field, and let x(t) be a solution of the differential
= 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
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
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