Solving a Homogeneous Linear Equation for x>0

[asdf1]

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

 

[LeonhardEuler]

Do you know what a Cauchy-Euler equation is?

 

[HallsofIvy]

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= x^r for some real number r?

 

[asdf1]

i don't know what a Cauchy-Euler equation is, but how'd you think of

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

 

[asdf1]

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?