1. Limited time only! Sign up for a free 30min personal 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!

Homogeneous Laplace's Equation

  1. Jun 27, 2010 #1
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution

    Using separable method I get
    Y"-kY= 0 and X"+kX=0

    For Case 1 and Case 2 where k>0 and k=0 there are no eigenvalues

    So Case 3 k<0 gives
    Y=ccos(sqrk x y) + dsin(sqrk x y)
    y=0, then c=0
    y=pie, then dsin(pie x sqrk)=0
    k= -n2, Yn=sin(ny)
    X"-n2X=0, X=ccosh(nx)+dsinh(nx)

    u(x,y)= summation (from n=1 to n=infinity) sin(ny)[cncosh(nx)+dnsinh(nx)]

    u(0,y)= summation (from n=1 to n=infinity) cnsin(ny)

    Using Fourier's Series to expand 0, I got 0 so cn is 0

    ux(x,y)= summation (from n=1 to n=infinity) sin(ny)[dnncosh(nx)]
    ux(x,0)= summation (from n=1 to n=infinity) sin(ny)[dnn]

    Using Fourier's Series to expand 3siny-5sin4y, I also got 0 so dn is 0 as well

    Therefore u(x,y)=0

    However the answer given in the book is

    u(x,y)=(3/cosh5)(siny)(sinhx) - (5/cosh20)(sin4y)(sinh4x)

    I realized that the book didn't used Fourier's Series but compared the coefficients of the two equations instead. But my question is what happened to the n because if we compare the coefficients aren't we supposed to get

    [3/ncosh(5n)] n=1 so we get [3/cosh(5)]
    [-5/nsinh(5n)] n=4 so we get [-5/4sinh(20)] so u(x,y) should be

    [3/cosh(5)](siny)(sinhx) - [5/4sinh(20)](sin4y)(sinh4x)

    Did I do it correctly or is it just a printing error?
  2. jcsd
  3. Jun 27, 2010 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    They left out a 4 multiplying the cosh(20) in the denominator
    And you meant cosh(20) instead of sinh(20). They just dropped a 4.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Homogeneous Laplace's Equation Date
Homogeneous Diff. Eqn Finding Solution Feb 17, 2018
Question ODE non-homogeneous Linear Oct 3, 2017
What does it mean for an equation to be homogeneous? May 31, 2017
Laplace of non-homogeneous Apr 23, 2011
Non-homogeneous laplace Jan 18, 2008