Results 1 to 4 of 4

Math Help - Most General solution of Laplace's equation

  1. #1
    Member Jason Bourne's Avatar
    Joined
    Nov 2007
    Posts
    132

    Most General solution of Laplace's equation

    Laplace's equation written in plane polar coordinates (r,\theta) is

    T_{rr} + \frac{1}{r}T_r + \frac{1}{r^2}T_{\theta \theta} = 0

    I need to find the most general solution T(r) which is a function of r only.

    Can anyone see what this would be?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,697
    Thanks
    1469
    Quote Originally Posted by Jason Bourne View Post
    Laplace's equation written in plane polar coordinates (r,\theta) is

    T_{rr} + \frac{1}{r}T_r + \frac{1}{r^2}T_{\theta \theta} = 0

    I need to find the most general solution T(r) which is a function of r only.

    Can anyone see what this would be?
    If T is a function of r only, then T_{\theta\theta}= 0 so the equation becomes the ordinary differential equation T_{rr}+ \frac{1}{r}T_r= 0. Let u= T_r and we have u_r+ \frac{1}{r}u= 0 which is a separable first order differential equation. \frac{du}{u}= -\frac{dr}{r}. Can you finish it from there?
    Last edited by HallsofIvy; January 16th 2009 at 08:34 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member Jason Bourne's Avatar
    Joined
    Nov 2007
    Posts
    132
    Quote Originally Posted by HallsofIvy View Post
    If T is a function of r only, then T_{\theta\theta}= 0 so the equation becomes the ordinary differential equation T_{rr}+ \frac{1}{r}T_r= 0. Let u= T_r and we have u_r+ \frac{1}{r}u= 0 which is a separable first order differential equation. \frac{du}{u}= -\frac{dr}{r}. Can you finish it from there?
    Thanks. So T = Aln(r) + B

    A,B constants?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,697
    Thanks
    1469
    Yes, that is correct.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. general solution of the equation
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: February 6th 2011, 12:40 PM
  2. Replies: 2
    Last Post: May 18th 2009, 12:51 PM
  3. General Solution of Differential Equation
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: December 7th 2008, 03:13 PM
  4. General Solution to an Equation
    Posted in the Calculus Forum
    Replies: 2
    Last Post: September 13th 2008, 04:56 AM
  5. General solution of this equation
    Posted in the Calculus Forum
    Replies: 5
    Last Post: April 4th 2008, 07:08 AM

Search Tags


/mathhelpforum @mathhelpforum