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Whoa! How did they do this ?

  1. Mar 21, 2008 #1
    [SOLVED] Whoa! How did they do this ?


    I am trying to teach myself partial diff equations and I'm stuck with this small step which I do not understand. I hope someone will help me.

    The problem statement, all variables and given/known data
    Solve the Partial Differential Equation

    Solution given in the book

    Subsidiary equations are


    The step which I do not understand is this :

    Using multipliers x, y, z. we get:
    each fraction=[tex]\frac{xdx+ydy+zdz}{x(x^2+y^2+z^2)}[/tex]

    Rest of the solution I understand but I don't get that how they did this particular step. How did they get this fraction ?

    Thanks for your time and effort:smile:
  2. jcsd
  3. Mar 21, 2008 #2


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    … ratios …

    Hi Google_Spider! :smile:

    It's easy algebra - but it looks extremely unlikely the first time you see it! :eek:

    If the ratio X/A = Y/B = a, then you can add any multiples together on top and bottom, and the ratio is the same:

    (pX + qY)/(pA + qB) = (paA + qaB)/(pA + qB) = a(pA + qB)/(pA + qB) = a.​

    And the same works for X/A = Y/B = Z/C = a, and so on.

    In this case, put p = x, q = y, r = z, and see what happens … :smile:
  4. Mar 21, 2008 #3
    Thanks tiny-tim!:smile:
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