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Solve Differential eq with initial conditions

  1. Oct 15, 2010 #1
    1. The problem statement, all variables and given/known data

    Solve: [tex]\frac{\partial H}{\partial t} = -4\kappa H^{2}[/tex]

    With initial condition: [tex]H(0) = 1/L^{2}[/tex]

    To find: [tex]H(t) = \frac{1}{4\kappa t + L^{2}}[/tex]

    2. The attempt at a solution

    I tried using Taylor series expansion such that:

    [tex]H(t)\approx H(0)+t\frac{\partial H}{\partial t}(0)+.....[/tex]

    To first order this yielded: [tex]H(t)=\frac{L^{2}-4\kappa t}{L^{4}}[/tex]

    This is wrong unless t and/or k equals zero. Therefore this is wrong, it is not the general solution. Please help if you can. Thanks.
     
  2. jcsd
  3. Oct 15, 2010 #2

    tiny-tim

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    hi billiards! :smile:
    eugh! :yuck:

    just separate the variables: dH/H2 = -4k dt :wink:
     
  4. Oct 15, 2010 #3
    Thanks tiny-tim, with that hint I solved it straight away. You have no idea how long I was stuck.

    Out of interest, what was wrong with my Taylor series approach?
     
  5. Oct 15, 2010 #4

    tiny-tim

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    he he :biggrin:
    nothing … they are the same to first order …

    what is the inverse of 1 + (4k/L2)t, to first order ? :wink:
     
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