I need to find a solution to:(adsbygoogle = window.adsbygoogle || []).push({});

[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.

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# DE Auxilary Equation

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