Differential Equations Solutions 114

Differential Equations Solutions 114 - 124 Chapter 20....

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Unformatted text preview: 124 Chapter 20. Solutions: Solution of Ordinary Differential Equations CHALLENGE 20.15. • Using the notation of the pointer, we let a(t) = 1 > 0, b(t) = 8.125π cot((1 + t)π/8), c(t) = π 2 > 0, and f (t) = −3π 2 . These are all smooth functions on [0, 1]. • Since c(t) = π 2 /2 > 0 and 1 [f (t)]2 dt = 0 π4 , 4 the solution exists and is unique. • Since f (t) < 0, the Maximum Principle tells us that max u(t) ≤ max(−2.0761, −2.2929, 0) = 0. t∈[0,1] • Letting v (t) = −3, we see that −v (t) + 8.125π cot((1 + t)π/8)v (t) + π 2 v (t) = −3π 2 and v (0) = v (1) = −3. Therefore the Monotonicity Theorem says that u(t) ≥ v (t) for t ∈ [0, 1]. • Therefore we conclude −3 ≤ u(t) ≤ 0 for t ∈ [0, 1]. Note on how I constructed the problem: The true solution to the problem is u(t) = cos((1 + t)π/8) − 3, which does indeed have the properties we proved about it. But we can obtain a lot of information about the solution (as illustrated in this problem) without ever evaluating it! CHALLENGE 20.16. Let y(1) (t) = a(t), y(2) (t) = a (t). Then our system is y (t) = y(2) (t) 2 y(1) (t) − 5y(2) (t) . function [t,y,z] = solvebvp() z = fzero(@fvalue,2); % {\tt now the solution can be obtained by using ode45 with % {\tt initial conditions [5,z].}}\STATE{\% {\tt For example, [t,y] = ode45(@yprime,[0:.1:1],[5 z]); % End of solvebvp ...
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This note was uploaded on 01/21/2012 for the course MAP 3302 taught by Professor Dr.robin during the Fall '11 term at University of Florida.

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