1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Can the 2nd ode x*y''-c*y=0 be solved exactly?

  1. Jul 2, 2010 #1
    As the tittle, can the 2nd ode (B.C not fixed, I need the general solution)

    y''-c/x*y=0

    be solved exactly? Or can it be translated into some
    special mathphysical equations, such as Bessel?
    Hypergeometric? etc

    any comments or references are welcome.

    Thank you for advance!
     
  2. jcsd
  3. Jul 3, 2010 #2
    Would this work?

    [tex]

    y'' - \frac{cy}{x} = 0 [/tex], can be rewritten as
    [tex]\frac{dydy}{y} = \frac{c}{x}dxdx[/tex], which upon integrating gives
    [tex]ln|y|dy = c*ln|x|dx[/tex], and then we would just have to integrate one more time?
     
  4. Jul 3, 2010 #3

    Mark44

    Staff: Mentor

    I would go with a series solution, with y = c0 + c1x + c2x2 + ... + cnxn + ...
     
  5. Jul 3, 2010 #4
    Don't you need to search for a solution not from the type:

    [tex]
    y(x)=\sum_{n=0}^{\infty}a_{n}x^{n}
    [/tex]

    but from the form:

    [tex]
    y(x)=\sum_{n=0}^{\infty}a_{n}x^{n+r}
    [/tex]

    [tex]
    a_{0}\neq 0
    [/tex]

    because of the singularity at x=0 ?

    I don't think it is possible to find both linearly independent series' if you search the solution in the first form, I couldn't. but may be I'm rusty in the Olde ODE's :)
     
  6. Jul 3, 2010 #5

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    No, that is not valid.
    [tex]\frac{d^2y}{dx^2}\ne \frac{(dy)(dy)}{(dx)(dx)}[/tex]
     
  7. Jul 3, 2010 #6

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

  8. Jul 3, 2010 #7

    Mark44

    Staff: Mentor

    The equation as originally given is xy'' - cy = 0, which is defined when x = 0. It is only by dividing by x that you introduce an singularity.
     
  9. Jul 3, 2010 #8
    Thans to all of you. Your browse or reply helps me.

    To this point, LCKurtz's message is the most nearest to
    my need!
     
  10. Jul 4, 2010 #9

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    No, this DE has a regular singular point at x = 0. gomunkul51 is correct about the form the infinite series would take. See

    http://banach.millersville.edu/~bob/math365/singular.pdf

    for the definition of a singular point and discussion.
     
  11. Jul 4, 2010 #10

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    You might also try a change of variables to transform the differential equation into the Bessel differential equation. For instance, u=sqrt(x) gets you a differential equation that sort of looks like the Bessel differential equation.
     
  12. Jul 4, 2010 #11

    Mark44

    Staff: Mentor

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook