Solve Inhomogeneous Problem Using Greens Function: (1-x2)y'' - xy' + y = f(x)

[sara_87]

Homework Statement

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

(1-x² ) y'' - x y' + y = f(x)

y(0) = y(1) = 0

Homework Equations

the answer is:

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

and -z(1-x² ) ^1/2 (1-z² ) ^1/2

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?

 

[MathematicalPhysicist]

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!