3 1. Notation and language 1.1. Numbers. We’ll write R for the set of all real numbers. We often think of the set of real numbers as the set of points on the number line. An interval is a subset I of the number line such that if a, c ∈ I, and b is a real number between a and c, then b ∈ I. There are several types of intervals, each with a special notation: (a, c) = {b ∈ R : a < b < c}, open intervals; (a, c] = { b ∈ R : a < b ≤ c} and [a, c) = { b ∈ R : a ≤ b < c}, half-open intervals; and [a, c] = {b ∈ R : a ≤ b ≤ c}, closed intervals. There are also unbounded intervals, (a, ∞) = {b ∈ R : a < b}, [a, ∞) = {b ∈ R : a ≤ b}, (−∞, c) = {b ∈ R : b < c}, (−∞, c] = {b ∈ R : b ≤ c }, and = R, the whole real line. The symbols ∞ and −∞ do not (−∞, ∞)represent real numbers. They are merely symbols so that −∞ < a is a true statement for every real number a, as is a < ∞. 1.2. D ependent and independent variables. Most of what we do will involve ordinary differential equations. This means that we will have only one independent variable. We may have several quant ities depending upon that one variable, and we may wish to represent them together as a vector-valued function. Differential equations arise from many sources, and the independent var ia ble can signify many different things. Nonetheless, very often it represents time, and the dependent variable is some dynamical quantity which depends upon time. For this reason, in these notes we will pretty systematically use t for the independent variable, and x for the dependent variable. Often we will write simply x, to denote the entire function. The symbols x and x(t) are synonymous, when t is regarded as a variable. We generally denote the deriva t ive with respect to t by a dot: dx x˙ = ,dt4 and reserve the prime for differentiation with respect to a spatial vari-able. Similarly, d2x ¨x = . dt2 1.3. Equations and Parametrizations. In analytic g eometry one learns how to pass back and f orth between a description of a set by means of an equation and by means of a para metrization. For example, the unit circle, that is, the circle with r adius 1 and center at the origin, is defined by the equation 2 2 x + y = 1 . xA solution of t his equation is a value of (x, y) which satisfies the equa-tion; the set of solutions of this equation is the unit circle. Any set will be t he solution set of many different equations; for example, this same circle is also the set of po ints (x, y) in the plane for which 4 + 2x2y2 + y4 = 1. This solution set is the same as the set parametrized by x = cos θ , y = sin θ , 0 ≤ θ < 2π . The set of solutions of the equation is the set of values of the parametriza-tion. The angle θ is the parameter which specifies a solution. An equation is a criterion, by which one can decide whether a point lies in the set or not. (2, 0) does not lie on the circle, because it doesn’t satisfy the equation, but (1, 0) does, because it does satisfy the equation. A parametrization is an enume ration, a listing, of all the elements of the set. Usually we try to list every element only once. Sometimes we only succeed in picking out some of the elements of the set; for example 2y = √1 − x , −1 ≤ x ≤ 1 picks out the upper semicircle. For emphasis we may say that some enumeration gives a complete parametrization if every element of the set in question is named; for example 2y = √1 − x , −1 ≤ x ≤ 1 , or y = x2 , −1 < x < 1 ,−√1 −is a complete parametrization of t he unit circle, different from the one given above in terms of cosine and sine. Usually the process of “ solving” and equation amounts to finding a parametrization for the set defined by the equation. You could call a5 parametrization of the solution set of an equation the “general solution” of the equation. This is the language used in Differential Equations. 1.4. Parametrizing the set of solutions of a differential equa-tion. A differential equation is a stated relationship between a function and its derivatives. A solution is a function satisfying this relationship. (We’ll emend this slightly at the end of this section.) For a very simple example, consider the differential equation ¨x = 0 . A solution is a function which satisfies the equation. It’s easy to write down many such functions: any function whose graph is a straight line satisfies this ODE. We can enumerate all such functions: they are x(t) = mt + b for m and b arbitrary real constants. This expression gives a parametriza-tion of the set of solutions of ¨x = 0. The constants m and b are the parameters. In our parametrization of the circle we could choose θ ar-bitrarily, and analog ously now we can choose m and b arbitrarily; for any choice, the function mt + b is a solution. Warning: If we fix m and b, say m = 1, b = 2, we have a specific line in the (t, x) plane, with equation x = t + 2. One can parametrize this line easily enough; f or example t itself serves as a para meter, so the points (t, t+2) run through the points on the line as t runs over all real numbers. This is a n entirely different issue fro m the parametrization of solutions of ¨ = 0. Be sure you understand this point. x 1.5. Solutions of ODEs. The basic existence and uniqueness theo-rem for ODEs is the following. Suppose that f(t, x) is continuous in the vicinity of a point (a, b). Then there exists a solution to ˙x = f(t, x) defined in some open interval containing a, and it’s unique provided ∂f /∂x exists. There ar e certainly subtleties here. But some things are obvious. The “uniqueness” part
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