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DE Auxilary Equation

  1. Oct 16, 2011 #1
    I need to find a solution to:
    [tex]x^{2}y"-xy'+y=0[/tex] in the form of [tex]y=x^{r}[/tex] where r is a constant.

    I started by finding the appropriate derivatives:
    [tex]y=x^{r}[/tex]
    [tex]y'=rx^{r-1}[/tex]
    [tex]y"=r^{2}x^{r-2}[/tex]

    Then substituting in:
    [tex]x^{2}(r^{2}x^{r-2})-x(rx^{r-1})+x^{r}=0[/tex]
    which simplifies to:
    [tex]r^{2}-r+1=0[/tex]

    I then solved and got the complex roots:
    [tex]\frac{1\pm i\sqrt{3}}{2}[/tex]

    I'm not sure what to do next. The examples I've seen so far have separated out the imaginary part using identities, where the function is exponential.
     
  2. jcsd
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