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Variation of Parameter Problem

  1. Feb 20, 2009 #1
    1. The problem statement, all variables and given/known data


    Find the general solution using the method of variation of parameters of:

    y''-6y'+9y=(x^-3)(e^3x)

    I found the roots of the corresponding homogeneous equation to be lamba = 3. So there are repeated roots. My question is, how do I solve a variation of parameter question with repeated roots? I know how to do it using reduction of order but confused on variation of parameters
     
  2. jcsd
  3. Feb 20, 2009 #2

    HallsofIvy

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    The same way you solve it without repeated roots! Since x= -3 is a double root of the characteristic equation, two independent solutions to the associated homogeneous equation are [itex]e^{-3x}[/itex] and [itex]xe^{-3x}[/itex].

    Look for a solution to the entire equation of the form [itex]y(x)= u(x)e^{-3x}+ v(x)xe^{-3x}[/itex]. Now just follow the usual procedure for variation of parameters.
     
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