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Differential equation with singular boundary conditions

  1. Apr 9, 2008 #1
    Hey guys, just need some hints with this doosey

    1. The problem statement, all variables and given/known data
    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 =)


    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
    Last edited: Apr 9, 2008
  2. jcsd
  3. Apr 9, 2008 #2


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    Science Advisor

    Well, using hint would be a good start! If you let u= y/x, so that y= xu, what does the equation become?
  4. Apr 9, 2008 #3
    With the substitution u=y/x equation becomes
    which has solution
    u=c1*cos(kx)+c2*sin(kx) [k=sqrt(a)]
    so that
    So doing y'(0)=c1=0 y(1)=c1*cos(k)+c2*sin(k)=0
    i.e. y=0, onlythe trivial solution ?
  5. Apr 9, 2008 #4
    Thats what i got (sorry if it wasnt obvious, yeh i used the hint =) )
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