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Bessel Differential Equation With Log

  1. Feb 20, 2010 #1
    I am trying to solved a differential equation of Bessel type,

    X^2 Y('')+XY(')+(X^2-n^2)Y+YlogX=0,

    where Y(')=d/dx.

    Please help me that how to deal with such equation.
     
  2. jcsd
  3. Feb 20, 2010 #2

    Astronuc

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    Last edited: Feb 20, 2010
  4. Feb 21, 2010 #3
    Thanks, Astrounus.
    actually Frobenius method don't work here because of the singularity of log(x), at x=1.
     
  5. Feb 21, 2010 #4
    Try to change independent variable from x to u = ln(x). The eq. simplified when I did it, although I haven't double-checked my result.

    Torquil
     
  6. Feb 21, 2010 #5
    Thanks Torquil.
    U know the original equation was,
    Y''+xy+n^2y+exp(-x)y=0,
    when i used the transformation x=exp(-z/2), i got
    z^2y''+zy'+(z^2-n^2)y+ylog(z)=0,
    which i already post,
    now according to u if z=log(u), than i am sure i will get my original equation (1), with exponential.
     
  7. Feb 21, 2010 #6

    Astronuc

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    That was my thinking, but does the equation have to apply at all x, or x=0? I believe the singularity of ln(x) or log(x) is at x=0.

    I've sometimes applied a Bessel's equation over a limited range, e.g., finite cylindrical or annular domain.
     
    Last edited: Feb 21, 2010
  8. Feb 21, 2010 #7
    Frobenius ought to work. solve the homogeneous bessels equation to get a complementary soltn usng frob technique, for non integral values of m. for integral vaues, it ought to give linearly dependent solns. and for a particular integral, try variation of parameters. or mayb greens fxn. i havent started dealing with bessels equation yet, so dunno if dis'l work.
     
  9. Feb 21, 2010 #8
    u r right the singularity is at x=0.
    from limit range u mean, to expand the log(x) into to its series form and than take first two or three terms.
    u know this problem is very interesting in the sense that it show the clash of linear and exponential behavior.
    if the DE is
    y''+(n^2+x)y=0, (1)
    its solution is Airy function, now if
    Y''+(n^2+exp(2))y=0, (2)
    its solution is Bessel function. now combining eq.(1) and (2)
    y''+(n^2+exp(x)+x)y=0 (3)
    eq(3) is the DE which i had posted.
     
  10. Feb 21, 2010 #9
    Frobenius don't work becuase of log(x) and its sigularity.
    from variation u mean chnage of variable, its also don't work because chnage of variable create image on the other term, like exp(x) have image of log(x) and vice versa.
    Green function can be helpful, can u explain it plz?
     
  11. Feb 22, 2010 #10
    sorry. turns out i was kinda wrong about everything else too. greens function is used to seek solutions of inhomogeneous equations. but isnt this one homogeneous? the coefficient of Y (i.e zeroth degree derivative) is (X^2 -m^2 + log X). da log term changes things completely. maybe an ordinary power series expansion abt an ordinary point? the functions regular over the entire finite domain cept at 0 (im pretty sure my suggestions are going from bad to worse)
     
  12. Feb 23, 2010 #11
    u can preempt the solution, as being possibly of the form f(x)Jn(x), instead of Jn(x). mayb work backwards. I hav no idea how the introduction of the logarithmic term is going to change the behaviour of the D.E. bt going by the nature of log fxns, possibly not uniformly.
     
  13. Feb 23, 2010 #12
    How about if i put
    y=exp(-x)J(x)
    hope it works.........
     
  14. Feb 23, 2010 #13
    interesting...lemme try
     
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