334_pdfsam_math 54 differential equation solutions odd

# 334_pdfsam_math 54 differential equation solutions odd -...

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Unformatted text preview: Chapter 5 x2 = 2 x3 = 0, x4 = 0, etc. Observe that x2 = 22 In general, xj = 2j Consequently, xn = 0 for n ≥ j . 11. (a) A general solution to equation (6) is given by x(t) = xh (t) + xp (t), where xh (t) = Ae−0.11t sin √ 9879t + φ k 2j (mod 1) = k (mod 1) = 0. 3 22 (mod 1) = 3 (mod 1) = 0. 1 (mod 1) = 1 (mod 1) = 0, 2 is the transient term (a general solution to the corresponding homogeneous equation) and xp (t) = 1 sin t + 0.22 √ 1 sin 2t + ψ , 1 + 2(0.22)2 tan ψ = − 1 √, 0.22 2 is the steady-state term (a particular solution to (6)). (xp (t) can be found, say, by applying formula (7), Section 4.12, and using Superposition Principle of Section 4.7.) Diﬀerentiating x(t) we get 1 v (t) = xh (t) + xp (t) = xh (t) + cos t + 0.22 √ 2 cos √ 2t + ψ . 1 + 2(0.22)2 The steady-state solution does not depend on initial values x0 and v0 ; these values aﬀect only constants A and φ in the transient part. But, as t → ∞, xh (t) and xh (t) tend to zero and so the values of x(t) and v (t) approach the values of xp (t) and xp (t), respectively. Thus the limit set of points (x(t), v (t)) is the same as that of (xp (t), xp (t)) which is independent of initial values. 330 ...
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## This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at University of California, Berkeley.

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