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Integrated Factors (DE)

  1. Feb 4, 2012 #1
    1. The problem statement, all variables and given/known data

    I have been dealing with Exact Equations in my DE class, and I came around this problem.

    (t^2-y^2)+(t^2-2ty)(dy/dt)=0

    This is obviously not an exact eqn. So I tried using integrated factors on it and try to find this "factor" μ.
    But no matter if I did it in terms of t or in terms of y, I couldn't separate it in terms of one variable.

    dμ/dt=(-2t)/(t^2-2ty)

    or

    dμ/dy=(2t)/(t^2-y^2)

    2. Relevant equations

    Is there any way that you can find an integrated factor which it is in terms of both variables?
    instead of t or y alone, both?



    3. The attempt at a solution

    I tried everything, and this topic is not even covered in class or in the book. I learnt this on my own and I have only learn Integrated factors in terms of y or in terms of t, not both.

    Help please.
     
  2. jcsd
  3. Feb 4, 2012 #2

    LCKurtz

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    A function ##M(x,y)## is said to be homogeneous of order n if ##M(\lambda x,\lambda y)= \lambda^nM(x,y)##. If ##M(x,y)## and ##N(x,y)## are both homogeneous of degree n, then the differential equation ##M(x,y)dx + N(x,y)dy = 0## can be converted to a separable DE with the substitution ##y=ux##.

    That applies to your question. Look at http://www.cliffsnotes.com/study_guide/First-Order-Homogeneous-Equations.topicArticleId-19736,articleId-19713.html [Broken] for a discussion of this type of equation.
     
    Last edited by a moderator: May 5, 2017
  4. Feb 5, 2012 #3
    Thanks, that just lighted a bulb in my head.
     
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