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Homework Help: Critical points (will be a function of beta and delta)

  1. Mar 8, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the critical points where [itex]\alpha[/itex] = 1.

    dv/dt = v2 + [itex]\alpha[/itex]v - u + [itex]\delta[/itex] [I will call this (1)]

    du/dt = [itex]\beta[/itex]v - u [call this (2)]

    For what values of [itex]\beta[/itex] and [itex]\delta[/itex] are there no critical points?

    2. Relevant equations

    3. The attempt at a solution

    So firstly I set (1) and (2) to = 0 and from (2) I get:
    u = [itex]\beta[/itex]v and now I subst. this back into (1)

    0 = v2 + v(1 - [itex]\beta[/itex]) + [itex]\delta[/itex]

    Now from the quadratic formula I have this

    v = {-(1 - [itex]\beta[/itex]) ± √[(1 - [itex]\beta[/itex])2 - 4[itex]\delta[/itex]]}/2

    Now here I am stuck and not sure how to simplify this or interpret this?
  2. jcsd
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