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Homogeneous differential equation

  1. Apr 20, 2010 #1

    annoymage

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    1. The problem statement, all variables and given/known data

    (1-xcotx)y''-xy'+y=0

    y1(x)=x is a solution

    find the second solution, y2(x), y1 and y2 are linear independent

    2. Relevant equations

    N/A

    3. The attempt at a solution

    i only know how to find it by auxiliary equation by substitute y=erx
    and also i cant use substitution y=xr

    how to do this?
     
    Last edited: Apr 20, 2010
    annoymage, Apr 20, 2010
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  3. Apr 20, 2010 #2

    gabbagabbahey

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    With these types of problems, where you are given one solution you usually look for a second solution of the form [itex]y_2(x)=u(x)y_1(x)[/itex]. So, try that substitution and see what you get.
     
    gabbagabbahey, Apr 20, 2010
  4. Apr 20, 2010 #3

    annoymage

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    y=ux

    y'=u'x+u

    y''=u''x+2u

    substitute and get

    -(x2u''+2xu)cot(x)-x2u'+xu''+2u=0

    like this right?

    but it seems to become more complicated is it? or should i continue?
     
    annoymage, Apr 20, 2010
  5. Apr 20, 2010 #4

    gabbagabbahey

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    You'll want to double check this :wink:
     
    gabbagabbahey, Apr 20, 2010
  6. Apr 20, 2010 #5

    annoymage

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    owho, yes yes, i get the answer. thank you very much.. ^^v
     
    annoymage, Apr 20, 2010
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