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Differential Equation: x^2y''-xy'-3y=2x^-(3/2)

  1. Jan 31, 2017 #1
    1. The problem statement, all variables and given/known data


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    3. The attempt at a solution
    I am not asking to find the answer, just wanted to know whether to use the variation of parameters or undetermined coefficients. Because this was on a test problem and I used variation of parameters instead. I know it is a Cauchy-Euler equation, but do you use the method of undetermined coefficients or variation of parameters?
     
  2. jcsd
  3. Feb 1, 2017 #2
    First, you want to put your equation in standard form as such:

    ##y''-\frac{1}{x}y'-\frac{3}{x^2}y=2x^{-\frac{7}{2}}##.

    Then observe that the terms on the left side decrease in power by an increment of one. This implies that you want solutions ##y## of a form where ##y## is a function of ##x## of some power ##k##, so that you can say that ##y'## is in the ##k-1## power. Then you have ##y''## in the ##k-2## power.

    For example, for ##y=ax^{k}##, ##y'=akx^{k-1}##, and ##y''=ak(k-1)x^{k-2}##.

    Plug these values in the differential equation, and you have:

    ##ak(k-1)x^{k-2}-\frac{1}{x}(akx^{k-1})-\frac{3}{x^2}ax^{k}=ax^{k-2}(k(k-1)-k-3)=2x^{-\frac{7}{2}}##.

    Now everything is in terms of the same power. I don't know how this would work for non-homogeneous equations, though. I don't exactly know what this method is called, in any case. But this is what I would have done.
     
    Last edited: Feb 1, 2017
  4. Feb 1, 2017 #3

    ehild

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    You can use both. Show your work.
     
  5. Feb 1, 2017 #4

    pasmith

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    Unless the question specifically tells you to use a particular method, any valid method should be acceptable.
     
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