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Reduction of Order not giving a second solution

  1. Mar 12, 2013 #1
    1. The problem statement, all variables and given/known data

    (x-1)y'' - xy' + y = 0, y=e^x is a solution

    2. Relevant equations



    3. The attempt at a solution

    Assume the second solution is of the form ve^x, where v' = (y^'2)e^-int[-x/(x-1)]

    So v' = e^(-2x)e^(x+ln|x-1|) = e^(ln|x+1|-x)

    Then, this second solution must be

    (e^x)(e^(ln|x-1|-x))

    =e^(ln|x-1)

    =x-1

    But, this is no solution to the DE. What went wrong?

    Thank you.



    NEVERMIND, I forgot to integrate v'. What was I thinking?
     
    Last edited: Mar 12, 2013
  2. jcsd
  3. Mar 13, 2013 #2

    Simon Bridge

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    You mean: $$y^{\prime\prime} - \frac{x}{x-1}y^\prime + \frac{1}{x-1}y = 0$$ $$\frac{d}{dx}\left ( v^\prime(x)e^{2x}e^{\int \frac{xdx}{x-1}} \right )=0$$
     
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