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Homework Help: First Order Linear Differential Equations

  1. Jan 30, 2008 #1
    1. The problem statement, all variables and given/known data

    In a particular cosmological model,
    the Friedmann equation takes the form L^2 (a')2 = a^2 − 2a^2 + 1, where L is a positive constant,
    the dot denotes time differentiation, and the initial condition is a(0) = 1. What are the units of
    L? Show, without solving this equation, that the universe described by this model is never smaller
    than a certain minimum size. Now solve the equation and describe the history of this universe.

    2. Relevant equations

    3. The attempt at a solution

    I basically considered this as an autonomous equation and found the critical points. Once A takes those values, the derivative will be 0 so the value of the function will not change. In class however, my tutor discussed some other weird (in my opinion) method of solving the problem which simply went over my head. Can someone please help me confirm whether I'm correct?

    Also, I don't see how we can "describe the history of this universe" by solving this equation. Please advise!!

    Thank-you very much for your kind co-operation!!
  2. jcsd
  3. Jan 30, 2008 #2


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    If I put a(0)=1 into your equation it looks like a'(0)=0. So the solution is the trivial solution a(t)=1 for all t. It hardly matters how you solve something like that. I suspect there is either a typo or unclear notation. Can you clarify what the ODE actually is??
    Last edited: Jan 30, 2008
  4. Jan 30, 2008 #3


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    Did you mean to write (a')^2? Also what is the point of writing a^2 − 2a^2 instead of −a^2?
  5. Jan 30, 2008 #4


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    Of course, the whole question makes no sense without saying what the variable a reoresents physically! The "diameter" of the universe?

    Also I notice this is titled "First Order Linear Differential Equations". While that differential equation is first order, it definitely is not linear!
  6. Jan 31, 2008 #5

    I agree with all of you but that's lecturer what it is. It's probably one of those questions which the lecturer set by mistake or just for the sake of it.

    Evidently, everything about it is wrong so I guess I'll just ignore the question. Thanks you guys for your help!!
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