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Differential Equations question

  1. Feb 25, 2012 #1
    1. The problem statement, all variables and given/known data
    Determine for which values of m the function ∅(x) = xm is a solution to the given equation
    a) 3x2y" + 11xy' -3y = 0
    b) x2 y" - xy' - 5y = 0


    3. The attempt at a solution
    I tried approaching this problem by substituting ∅(x) into the question.
    a) 3x2(xm)'' + 11x(xm)' - 3(xm) = 0
    xm(3x2m2 + 11xm - 3) = 0
    I don't know what to do after this step and I'm not sure this is the correct way to do this problem. I just tried doing this because I looked at a similar question and this is how it was approached in the solutions manual.
     
  2. jcsd
  3. Feb 25, 2012 #2
    Check your differentiation. What is d2/dx2(xm)?
     
  4. Feb 25, 2012 #3
    y'= mxm-1 y''=(m-1)(m)xm-2 Are you asking me to substitute these into the given equation? I actually did that but what am I going to do with the x variable since i'm looking for what m is equal to?
     
  5. Feb 25, 2012 #4
    Yes. Substitute them back in, but mind how the powers of x add up. What do you get?
     
  6. Feb 25, 2012 #5
    3x2(mxm-1) + 11x(m-1)(m)xm-2 - 3(xm) = 0
    3x2mxm-1 + 11x(m2-m)xm-2 - 3xm = 0
    xm(3x2mx-1 + 11x(m2-m)x-2 - 3) = 0
    m3x + 11x - 1(m2-m) - 3 = 0
    m3x + m211x-1 - m11x-1 - 3 = 0
    m(3x + m(11/x) - 11/x) = 3

    I stopped right there since I wasn't really sure how I was going to get rid of the x...
     
  7. Feb 25, 2012 #6
    Shouldn't it be 3x2((m-1)(m)xm-2) + 11x(mxm-1) - 3(xm) = 0?
     
  8. Feb 25, 2012 #7
    oh my god... I can't believe I didn't catch that. Okay, let me try to do this again.
     
  9. Feb 25, 2012 #8
    I finally got the answer. I can't believe one stupid mistake like that got me stuck on this problem. Thanks a lot for pointing out that mistake for me, alan!
     
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