Basis question

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asdf1
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Can someone give suggestions for this question?

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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  • #2
LeonhardEuler
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Do you know what a Cauchy-Euler equation is?
 
  • #3
HallsofIvy
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Also called an "equi-potential" equation since the "power" of x is always "equal" to the order of the derivative.

What happens if you try a solution of the form y= xr for some real number r?
 
  • #4
asdf1
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i don't know what a Cauchy-Euler equation is, but how'd you think of
HallsofIvy said:
.
What happens if you try a solution of the form y= xr for some real number r?
 
  • #5
asdf1
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ok, I've taken a look at the Cauchy-Euler equation in the AEM textbook, but there's two things that i think are strange:

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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