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Solving a differential equation using Cauchy-Euler Method

  1. Mar 24, 2012 #1
    1. The problem statement, all variables and given/known data
    Hey all, this is my first time posting here. I've used your help before, but have not actually ever had to post a question. Thanks for any help you can give, I am excited to finally join the community.

    I am doing calculus homework and I am having trouble solving this problem using the Cauchy-Euler method that we're supposed to solve it with.

    Solve: xy4 + 6ym = 0

    It is the part about the 6ym that I am not sure what to do with.


    2. Relevant equations

    andny / dxn ....a0y = 0

    3. The attempt at a solution

    My book toldme to try and substitute y=xm into the derivative parts, and all of the other problems I've done work with that method. This is what I've done so far.

    y=xm
    y1=mxm-1
    y2=m(m-1)xm-2
    y3=m(m-1)(m-2)xm-3
    y4=m(m-1)(m-2)(m-3)xm-4

    I know how to treat the 4th derivative term; again, I just don't know what or how to think about the y^m term. I know the solution to the problem, as it is in the back of my book, I just don't know how to get there. Can anyone give me a prod in the right direction?
    Thanks
     
  2. jcsd
  3. Mar 24, 2012 #2

    HallsofIvy

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    I don't see any derivative in the original equation!
     
  4. Mar 24, 2012 #3
    I'm sorry, our book uses notation so that y^4 is supposed to be the fourth derivative of y.
    So, for example, y1=y´ and y^2 = y´´
     
  5. Mar 24, 2012 #4

    Ray Vickson

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    The usual notation is y^(2) (so we can tell the difference between that and the square of y). Anyway, you have found the first few y^(m) for y = x^n. If you don't see the pattern, I suggest you look instead at y^(m)/m! and see if you recognize the coefficients there.

    RGV
     
  6. Mar 24, 2012 #5
    I see that there is a relationship in that
    y^(m)/m = (my^(m-1) + m(m-1)y^(m-2)....)/(m(m-1)(m-2)...) but I don't actually understand how to apply that relationship at all yet
     
  7. Mar 24, 2012 #6
    Nevermind, I'm really sorry about this. I mistook the book for having written y^m when they really wrote y^´´´. I guess my new problem is poor eyesight. Thanks a lot for the help, I struggled trying to solve this for a solid 40 minutes.
     
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