334_pdfsam_math 54 differential equation solutions odd

334_pdfsam_math 54 differential equation solutions odd -...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

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.

Ask a homework question - tutors are online