- 300 12 Differential Equations Proof Let Y F X Dx If Y 0 Then The Inequality Is Trivial Otherwise Using The Sta 1 (115.78 KiB) Viewed 59 times
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300 12 Differential Equations PROOF. Let y = f(x) dx. If y = 0, then the inequality is trivial. Otherwise, using the standard inner product on R", 11? = </* F(x) dx, y) – L*P(),y) dx = [" (F%),9% de sfi()1 yilda = 131 Flul F(x1)|dx. The second line follows by the Schwarz inequality. Dividing through by ly gives the result. Exercises for Section 12.2 A. If/:.,- is differentiable atxe 14,6], show that is continuous atxo. 1. Show thet it c, are real members and are Riemann integrable functions from (a, b) to R"then af+Bg is Riemann integrable and Said: +8 C. Derive Theorem 12.2.6 from the one-variuble version, D. Prove that every continuous function : a. b) +R" is Riemann integrable. E. Define a regular curve to be a differentiable function S: 19.) " such that "(x) is never the zero vector (a) Given two regular curves f: 9.b - and g: cd-R", we say that g is a repara- metrization of fif there is a differentiable l'unction hd -- [a, b] such that # (1) #0 for all t and g = foh. Show that g'(d) = f'(20)W). (6) Define the length of a regular curveſ to be L) FOX)||dx. Show that the length is not affected by reparametrization. HINT: Consider reparametrizations where w) is always positive or always negative. (c) Given a regular curve / show there is a reparametrization g with 18 = 1 for all! Such a curve has imit speed. HINT: Show thatx finde has an inverse function, 12.3 Differential Equations and Fixed Points The goal of this section is to start with a DE of order n, and convert it to the problem of finding a fixed point of an associated integral operator. The first step is to take a fairly general form of a higher-order differential equation and turn it into a first-order DE at the expense of making the function vector-valued. We define an initial value problem, for functions on (0,6) and a point ce [a,b], as f)(x) = (.*./(x). S'(x)...../N-)(x)), (12.3.1) f(e) = f'(0) - M. ...pr-1)(0) = *-1, (12.3.2) where is a real-valued continuous function on ſa, 6 x 7". This is not quite the most general situation, but it includes most important examples. The first equation