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Finding general solution of an ode using substitution

  1. Nov 25, 2015 #1
    1. The problem statement, all variables and given/known data

    By making the transformation u= x^αy where α is a constant to be found, find the general solution of


    y'' + (2/x)y' + 9y=0


    3. The attempt at a solution

    I've worked out y,y',y'' and subbed them in to get

    x^-au'' + x^a-1(2-2a)u' + x^-a-2(x^2-a(a-1))u =0

    but I don't know what to do from here.

    Any help would be greatly appreciated.
     
  2. jcsd
  3. Nov 25, 2015 #2

    Orodruin

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    Can you perhaps chose ##a## in some intelligent fashion to simplify your equation?
     
  4. Nov 25, 2015 #3
    yes sorry I forgot that bit I had worked out that a should be -1 and there's a typo it should be a(a+1)
    which would simplify it down to

    x^1 U'' + x^1U=0

    then I had the thought to divide by x to get U''+U=0

    but I got stuck here because I wasn't sure on what to do. Should I change back to y?
     
  5. Nov 25, 2015 #4

    Orodruin

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    No, you should solve the differential equation you obtained for u. Reinsering y would just give back the old differential equation.

    Also, your original equation had a 9. Where did that go?
     
  6. Nov 25, 2015 #5
    OK so I would have U''+9U=0

    with general solution of U=Ae^3x+Be^-3x

    and U=y/x so y=(Ae^3x+be^-3x)/x
     
  7. Nov 25, 2015 #6

    Orodruin

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    No, this does not solve the differential equation.
     
  8. Nov 25, 2015 #7
    sorry U should be Acos3x+Bcos3x
     
  9. Nov 25, 2015 #8

    Orodruin

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    Correct. For equations of this type in general, you may want to have a look at spherical Bessel functions.
     
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