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Partial differential equations , rearranging and spotting

  1. Aug 27, 2011 #1
    I have been given an equation : r^2 ( d^2R/dr^2) + 2r(dR/dr) - lambda*R = 0


    It says to assume R~ r^β



    Then i can't seem to spot how from that information we can produce this equation:

    β(β − 1) rβ + 2β rβ − λ rβ = 0


    Any help would be appreciated, thanks.
     
  2. jcsd
  3. Aug 27, 2011 #2

    Hootenanny

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    Simply substitute [itex]R=r^\beta[/itex] into the differential equation.
     
  4. Aug 27, 2011 #3

    HallsofIvy

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    That is, by the way, an "Euler-Lagrange" type equation. Each derivative is multiplied by a power of x equal to the order of the derivative. The substitution t= ln(r) changes it to a "constant coefficients" problem. You should remember that for such an equation we "try" a solution of the form [itex]e^{\beta t}[/itex] (although we then find that there are other solutions). With t= ln r, that becomes [itex]e^{\beta ln(r)}= e^{ln r^\beta}= r^\beta[/itex].
     
  5. Aug 27, 2011 #4
    Set R= constant * r**beta so dR=constant * whatever

    They are proportional
    R=k*r--beta , all k's ( constants cancel )
     
  6. Aug 27, 2011 #5

    Hootenanny

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    Why the constant?
     
  7. Aug 27, 2011 #6
    Well because I thought tilde is for (constant) linear proportionality.
     
  8. Aug 29, 2011 #7
    I have been given an equation
     
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