Differential equation with singular boundary conditions

[FunkyDwarf]

Hey guys, just need some hints with this doosey

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.

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 I am 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

 

[HallsofIvy]

Well, using hint would be a good start! If you let u= y/x, so that y= xu, what does the equation become?

 

[Roberto Bramb]

With the substitution u=y/x equation becomes
u''+au=0
which has solution
u=c1*cos(kx)+c2*sin(kx) [k=sqrt(a)]
so that
y=xu=c1*x*cos(kx)+c2*x*sin(kx)
So doing y'(0)=c1=0 y(1)=c1*cos(k)+c2*sin(k)=0
i.e. y=0, onlythe trivial solution ?

 

[FunkyDwarf]

Thats what i got (sorry if it wasnt obvious, yeh i used the hint =) )