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Math Help - 2nd linearly independent solution using reduction of order

  1. #1
    Member diddledabble's Avatar
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    Jul 2009
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    Post 2nd linearly independent solution using reduction of order

    How do you find the second linearly independent solution using reduction of order.

    Examples
    <br /> <br />
t^{2}y'' + 6ty' +6y = 0 <br />
t > 0<br />
f(t) = t^{-2}<br />

    <br /> <br />
ty'' + (1-2t)y' + (t-1)y = 0<br />
t > 0<br />
f(t) = e^t<br />
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  2. #2
    Senior Member Sampras's Avatar
    Joined
    May 2009
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    Suppose we have  t^{2}y'' + 6ty' +6y = 0, \ t > 0, \ y_{1}(t) = t^{-2}.

    We know that the second solution will have the form  y_{2}(t) = v(t)y_{1}(t) . Now:

     y_{2}(t) = t^{-2}v

     y_{2}'(t) = -2t^{-3}v+t^{-2}v'

     y_{2}''(t) = \cdots

    Plug these into differential equation. You will get an expression only involving derivatives of  v . Then make a substitution to get a linear first order differential equation.
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