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Homework Help: Greens function

  1. Apr 6, 2010 #1
    1. The problem statement, all variables and given/known data

    use the greens function G(x,z) to solve inhomogeneous problem:

    (1-x2 ) y'' - x y' + y = f(x)

    y(0) = y(1) = 0

    2. Relevant equations

    the answer is:

    G(x,z)= -x for x<z

    and -z(1-x2 ) 1/2 (1-z2 ) 1/2


    3. The attempt at a solution

    the general solution to the equation

    (1-x2 ) y'' - x y' + y = 0

    is:

    y = Ax + B(1-x2 ) 1/2



    i found B(z) and D(z) = 0 after subing in x= 1 and x= 0

    then i got:

    G(x,z) = A(z)

    and D(z)(1-x2 )1/2

    then i did this:

    -D(z)x(1-x2) -1/2 - A(z) = 1 *

    D(z)(1-x2) 1/2 - A(z)x = 0 **

    now we have 2 equation with 2 unknown which i can solve... but i didn't get the right answer so i just need to check are the 2 equations * and ** correct???
     
  2. jcsd
  3. Apr 9, 2010 #2

    MathematicalPhysicist

    User Avatar
    Gold Member

    Well, you forgot something it should be:
    G(x,z)=A(z)x for 0<=x<z<=1
    D(z)(1-x^2)^0.5 for 0<=z<x<=1
    And another thing, you should multiply the homogenous equation by what is called an integration factor, i.e we want the euqation to be in form: d/dx [p(x)d/dx y]+q(x)y=0
    so here if u is this factor then: u(1-x^2)=p and -ux=p' p'/p=x/(x^2-1) ln(p)=ln(x^2-1)/2
    so p=sqrt|1-x^2|, now as it's written in wiki, the difference in the derivative of dG(x,z)/dx for x=z from right minus dG(x,z)/dx for x=z from left equals 1/p(z), and also you have the property of symmetry of G, i.e G(x,z)=G(z,x).

    I hope I helped you some how, cheers!
     
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