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Ordinary differential equations

  1. Sep 12, 2010 #1
    Find equilibrium solutions for the following ODE initial-value
    problem and linearize the problem about those solutions
    z" = z − z^3,
    z(0) = z0
    z'(0) = v0

    i found the equilibrium solutions to be 0,1,-1. what are the steps to linearize the problem around these?
     
  2. jcsd
  3. Sep 12, 2010 #2
    I believe you represent it as a coupled system of first-order ODEs, then translate the equilibrium point to the origin via the transformations u=x-x_0 and v=y-y_0. Then compute the Jacobian at the equilibrium point and thus convert the system to a linear system with an equilibrium point at the origin. Try and find "Differential Equations" by Blanchard, Devaney, and Hall. It's an easy read and does a lot of work with systems.
     
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