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

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