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A Second Order Differential equation Bessel-type

  1. Sep 14, 2016 #1
    Hello!

    Im trying to solve this second order differential equation:

    \begin{equation*}
    -\dfrac{d^2y}{dx^2}+\dfrac{3}{x}\dfrac{dy}{dx}+(x^2+gx^4+2)y=0
    \end{equation*}

    Any idea?

    Maybe it could be converted to a Bessel-like equation (?) with an appropriate change of variables.

    The equation arises when your are considering a -2 dimensional (yes!, its correct: "Negative dimension") anhamonic oscillator.

    Thanks!
     
  2. jcsd
  3. Sep 15, 2016 #2

    Ssnow

    User Avatar
    Gold Member

    Hi, there is a proposition that says, if you have an equation of this kind:

    ## y''+P(x)y'+Q(x)y=0##
    where ##P## and ##Q## continuous on ##I## and if ##\lambda(x)## is a particular solution with ##\lambda(x)\not=0##, then the general solution is:

    ##y(x)=\lambda(x)\left(c_{1}+c_{2}\int\frac{e^{-\int P(x)dx}}{\lambda(x)^2}dx\right)##

    where ##c_{1},c_{2}## are constants.

    So you can start to search a particular solution and then apply this...
     
  4. Sep 16, 2016 #3
    I not sure how helpful this will be, but I think you're equation is a general confluent hypergeometric differential equation: http://mathworld.wolfram.com/GeneralConfluentHypergeometricDifferentialEquation.html.

    The general solutions are:

    [itex] y_1 = x^{-A} e^{-m\left(x\right)} F\left(a;b; h\left(x\right)\right) [/itex]
    [itex] y_2 = x^{-A} e^{-m\left(x\right)} U\left(a;b; h\left(x\right)\right) [/itex]

    where [itex] F\left(a;b;h\left(x\right)\right) [/itex] and [itex] U\left(a;b;h\left(x\right)\right) [/itex] are confluent hypergeometric functions of the first and second kind.
    Admittedly it can be challenging to identify the functions [itex] m\left(x\right) [/itex] and [itex] h\left(x\right) [/itex] . I'd try a few simple functions to see if you can make it work.

    Alternatively what is the constant g? Is it a small number? Can you use a perturbative expansion to find an approximate solution?
     
  5. Sep 18, 2016 #4
    Variation of parameters! I'll give it a try!

    Thanks
     
  6. Sep 18, 2016 #5
    Thanks I will try with this "factorization" procedure.
     
  7. Oct 21, 2016 #6
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