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Homework Help: Converting 2nd order ODE to Bessel Function

  1. Sep 7, 2008 #1
    1. The problem statement, all variables and given/known data
    I am attempting to solve the 2nd order ODE as follows using the generalized solution to the Bessel's equation


    2. Relevant equations
    original ODE:
    x[tex]d^{2}y/dx^{2}[/tex]-3[tex]dy/dx[/tex]+xy=0

    3. The attempt at a solution
    My first thought is to bring out an x^-1 outside of the function so that I end up with:
    [tex]x^{-1}([/tex][tex]x^{2}[/tex][tex]d^{2}y/dx^{2}[/tex]-3x[tex]dy/dx[/tex]+[tex]x^{2}[/tex]y)=0
    I would then solve the resulting Bessel equation found inside the parentheses, and multiply the resulting solution by x^-1. Is this at all a legal operation? Thank you.
     
  2. jcsd
  3. Sep 7, 2008 #2

    HallsofIvy

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    There is no need for the x-1 outside. Just multiply both sides of the original equation by x.
     
  4. Sep 7, 2008 #3
    Now if the opposite were true and I was trying to drop the power of x's by 1:

    example: (x^3)[tex]d^{2}y[/tex]/[tex]dx^{2}[/tex]+(x^2)dy/dx+(x^3)y=0

    Could I then instead divide by x to come up with:
    (x^2)[tex]d^{2}y[/tex]/[tex]dx^{2}[/tex]+(x)dy/dx+(x^2)y=0

    with the understanding that the solution to the resulting Bessel function would exclude any results for when x-> 0?
     
  5. Sep 7, 2008 #4

    HallsofIvy

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    Well, because x2 still multiplying the second derivative that would be a problem any way, but you are right if you divided by something that completely got rid of a function multiplying the highest derivative, then you would have to add that condition.
     
  6. Sep 23, 2010 #5
    Hey I need some help in converting the second order differential equation into..
    I was able to convert the original equation into the following form x^2*y''+2x*y'+x^2*y=0
    I am not able to move forward from here..
    Please could you suggest some method
     
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