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A model problem about singular limits

  1. Sep 16, 2011 #1
    The problem statement, all variables and given/known data

    Consider the linear differential equation εx'' + x' + x = 0, subject to the initial conditions x(0) = 1, x'(0) = 0.

    a) Solve the problem analytically for all ε > 0.
    b) Now suppose ε≪1. Show that there are two widely separated time scales in the problem, and estimate them in terms of ε.
    c) Graph the solution x(t) for ε≪1, and indicate the two time scales on the graph.
    d) What do you conclude about the validity of replacing εx'' + x' + x = 0 with its singular limit x' + x =0?
    e) Give two physical analogs of this problem, one involving a mechanical system, and another involving an electrical circuit. In each case, find the dimensionless combination of parameters corresponding to ε, and state the physical meaning of the limit ε≪1.

    The attempt at a solution

    The corresponding characteristic equation is ελ^2 + λ +1 = 0 and λ = (-1 ± √1 - 4ε)/2ε. Since ε > 0 we have 1 - 4ε < 0 and two complex roots.
    So x = Ae^(r*t)*cos(ω*t) + Be^(r*t)*sin(ω*t), where r = -1/2ε and ω = (|√1-4ε|)/2ε.
    Using x(0) = 1 and x'(0) we get A = 1 and B = - r/ω.
    Thus x = e^(r*t)*cos(ω*t) - (r/ω)e^(r*t)*sin(ω*t).
  2. jcsd
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