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Find a basis of solutions for the following second-order homogeneous linear equation for positive x:

x^2y``-xy`+y=0

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- #1

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Find a basis of solutions for the following second-order homogeneous linear equation for positive x:

x^2y``-xy`+y=0

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LeonhardEuler

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Do you know what a Cauchy-Euler equation is?

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HallsofIvy

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What happens if you try a solution of the form y= x

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HallsofIvy said:.

What happens if you try a solution of the form y= x^{r}for some real number r?

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1) for the case of double roots, the proof to the general solution only considers

x>O... why?

2) for the case of complex conjugate roots, there's only a general solution for all

positive x... why?

btw, i've been thinking all day of y=x^r... i'm still stumped over how'd you think of that?

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