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Hey guys, just need some hints with this doosey

We have

(x^2 y')' + ax^2y = 0 where a the eigenvalue (a sturm-lioville problem) (sp?)

with y'(0)=y(1) = 0 and we get the hint to substitute f = y/x.

Ok so i get the general solution being a sum of cosines and sines and i fiddled around a LOT with limits and exponentials and small angle approximations etc to try and get around the fact that the function blows up at x = 0 and were supposed to fit a boundary condition there. In all my efforts, the ones that make SOME sense, i get the eigen values = 0 which kind of makes it a trivial problem. I also tried taking advantage of the orthogonality of solutions given we know the weighting function x^2 but that required me to know what at least one eigenvalue was, and when i tried zero i had trouble solving exactly for the second but graphically it appeared to be close to root 2 but not quite.

Im sure im missing something stupid so if you could, gently, point that out that would be super =)

Cheers

-Z

EDIT: i also tried a series expansion with the first 3 terms for sine and cos but got lost =( i thought about trying a general series solution but wasnt sure

**1. Homework Statement**We have

(x^2 y')' + ax^2y = 0 where a the eigenvalue (a sturm-lioville problem) (sp?)

with y'(0)=y(1) = 0 and we get the hint to substitute f = y/x.

**3. The Attempt at a Solution**Ok so i get the general solution being a sum of cosines and sines and i fiddled around a LOT with limits and exponentials and small angle approximations etc to try and get around the fact that the function blows up at x = 0 and were supposed to fit a boundary condition there. In all my efforts, the ones that make SOME sense, i get the eigen values = 0 which kind of makes it a trivial problem. I also tried taking advantage of the orthogonality of solutions given we know the weighting function x^2 but that required me to know what at least one eigenvalue was, and when i tried zero i had trouble solving exactly for the second but graphically it appeared to be close to root 2 but not quite.

Im sure im missing something stupid so if you could, gently, point that out that would be super =)

Cheers

-Z

EDIT: i also tried a series expansion with the first 3 terms for sine and cos but got lost =( i thought about trying a general series solution but wasnt sure

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