Finding a basis of solutions for (x^2)y'' - xy' + y = 0

[asdf1]

for the following question:
find a basis of solutions for
(x^2)y`` -xy`+y=0


my problem:
i haven't a clue on where to start~
can someone give some tips for starters?

 

[Tide]

Have you learned power series methods (Frobenius) for solving linear ODEs?

 

[asdf1]

no, not yet

 

[HallsofIvy]

It's not necessary to use power expansions here. This is an "Euler-type" or "equipotential" equation- the degree of x in each term is equal to the order of the derivative.

Hint- let y= x^r for unknown r, then find r so that the equation is satisfied.

(That will get you started, but that's actually the easy part!)

 

[asdf1]

ok, i'll try that!