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Second Order ODE Homogenous

  1. Oct 22, 2015 #1
    1. The problem statement, all variables and given/known data
    (Reduction of order) The function y1 = x-1/2cosx is one solution to the differential equation x2y" + xy' + (x2 - 1/4) = 0. Use the method of reduction of order to find another linearly independent solution.

    3. The attempt at a solution

    I divided x2 to both sides to get the equation into y" + py' + qy = 0

    y" + (1/x)y' + ((1 - (1/4x2)) = 0

    Using Abel's method

    c e-∫(p)dx

    = c e-ln x
    y2 = -c lnx


    Am I doing this right?
     
  2. jcsd
  3. Oct 23, 2015 #2

    SteamKing

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    Take y2 and plug it back into the original differential equation and see if it is satisfied. :wink:
     
  4. Oct 23, 2015 #3

    vela

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    No, the problem said to use the method of reduction of order.

    But first, did you type the differential equation and solution correctly? I ask because the supposed solution doesn't satisfy the given differential equation.
     
  5. Oct 23, 2015 #4

    HallsofIvy

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    You mean x2y" + xy' + (x2 - 1/4)y = 0, don't you? As Vela pointed out, the given y1 does not satisfy the equation you gave. Perhaps it satisfies this one. I haven't checked.

    So you start by looking for a solution of the form [itex]y_2= u(x)x^{1/2}cos(x)[/itex]
    Find the first and second derivatives of that and put them into the equation. If your given function really is a solution to the differential equation, then all terms involving only "u" (as opposed to u' or u'') will cancel leaving a first order equation for v= u'.

     
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