Help me please

  1. Help me please!!!!!!!

    Can you help me to solve this:

    (d² e(r)/dr²)+(1/r)*(d e(r)/dr)=0

    There is no initials conditions, please use general form
  2. jcsd
  3. arildno

    arildno 11,265
    Science Advisor
    Homework Helper
    Gold Member

    Convince yourself of the following equality:
  4. Yes, but....

    And, if i want to find the general form of e(r)?
  5. arildno

    arildno 11,265
    Science Advisor
    Homework Helper
    Gold Member

    You get:
    This differential equation can be directly integrated to find the general solution.
    Last edited: May 23, 2004
  6. Maybe this?

    If I understand, you've made:

    If i integrate, i find:

    Wrong or not ?
  7. Thanks

    THANKS!!!!! you save me!!!
  8. I would also do it this way:



    I would then multiply through by r^2:


    I would recognize this as a d.e. of the Euler-Cauchy form:

    [tex]x^2y''+axy' + by=0[/tex]

    In the case of the given equation, a=1 and b=0. The characteristic equation for the Euler-Cauchy is:


    In our case:


    For the case of a real double root in the characteristic equation, the general solution for the Euler-Caucy is given as:

    [tex]y=(A + B\ln x)x^m[/tex]

    So in our case:

    e(r)&=(A + B\ln r)x^0\\
    e(r)&=A + B\ln r

    I guess this solution depends on having the Euler-Cauchy form available to you in your course, which may not be the case.
  9. HallsofIvy

    HallsofIvy 41,270
    Staff Emeritus
    Science Advisor

    Another way to do this problem is let u= e' so that u'= e" and the equation reduces to the separable first order equation u'+ (1/r)u= 0. Then du/u= -dr/r and so
    ln(u)= -ln(r)+ C1 or u= e'= C1/r. Integrating again, e= C1ln|r|+ C2.
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