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Homework Help: Seperaion of Variables (PDE's)

  1. Aug 8, 2010 #1
    1. The problem statement, all variables and given/known data
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    Look for a seperable solution T(r,θ) = R(r)Θ(θ) and derive equations for R(r) and Θ(θ) choosing a seperation constant that gives sinusoidal solutions for Θ(θ). Write down a general solution for Θ(θ) and show the equation for R(r) has solutions of the form R(r)=r^p.

    2. Relevant equations
    p is just a random letter which has no real meaning.


    3. The attempt at a solution
    I can get the solution of Θ(θ)=Asinkθ+Bcoskθ using k^2 as the seperation constant. That leaves me with r2R''+ rR' = k2R but im not sure what solution that gives, likewise for the general solution for Θ(θ). Any help will be much appreciated. Thanks
     
    Last edited: Aug 8, 2010
  2. jcsd
  3. Aug 8, 2010 #2
    When you are required to "show" a function satisfies a DE, then just substitute that function into the DE and see if it satisfies it. So you're asked to "show" that R(r)=r^p satisfies the DE:

    [tex]r^2 R''+rR'=k^2 R[/tex]

    so when you substitute R(r)=r^p into that, what must the relationship between p and k be so that it satisfies the DE?
     
  4. Aug 8, 2010 #3
    [itex]r^2 R''+rR' - k^2 R= 0[/itex] is Euler equation and solution is [itex]r^n \hbox { or } r^p[/itex] what ever which way you want to call it.
     
  5. Aug 11, 2010 #4
    okay thanks for clearing that up, but what about the general solution for Θ(θ)? still not sure on that...
     
  6. Aug 11, 2010 #5

    vela

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    You said you already found it in your original post. What specifically are you stuck on?
     
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