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Differential Equation: Separation of Variables

  1. Sep 4, 2013 #1
    1. The problem statement, all variables and given/known data
    dL/dp = L/2, L(0) = 100. Find the solution to the differential equation, subject to the given initial condition. My textbook says the answer is L = 100ep/2, but I don't know how to get that answer (or e for that matter).

    2. Relevant equations
    ?


    3. The attempt at a solution
    dL/dp = L/2
    L dL/dp = 1/2
    ∫LdL = ∫(1/2)dp
    L2/2 = x/2 + C
    L2 = x + 2C
    (0)2 = 100 + 2C

    Thanks for any help ahead of time!
     
  2. jcsd
  3. Sep 4, 2013 #2
    I found two mistakes in your procedure. In the second step of your solution, the L on the left side should divide dL.

    Then, the way to apply the initial condition is to make x=0, L=100, and solve for C. You did x=100 and L=0. D:
     
  4. Sep 4, 2013 #3
    Oh, man. D: Thanks for pointing that out!
     
  5. Sep 4, 2013 #4
    Now that I think about it, it looks like I'm supposed to use dy/dx = ky instead of what I did in my attempt?
     
  6. Sep 4, 2013 #5

    Dick

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    You don't need a formula, you can derive the result by separation of variables. But as ymlc pointed out you are making a mistake. You should have dL/L=(1/2)dp after separating. NOT LdL=(1/2)dp. Work from there.
     
  7. Sep 4, 2013 #6

    LCKurtz

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    That equation can be solved either by separation of variables or using linear equation theory and multiplying by an integrating factor. Or, for that matter, as a constant coefficient DE if you have had that method.
     
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