Dear MHF members,
I need help in finding the Green's function for the BVP
with and .
Many thanks.
bkarpuz
Note. Typo is corrected after TheEmptySet's warning.
I am going to assume that there is a typo in the statement of the problem otherwise the boundary condition at zero isn't needed as the solution to the homogenous problem is
Which is bounded no matter what the constants are but if we solve the problem
Now this is in its self adjoint form.
Solving the homogeneous equation gives
Now we need to find two solutions each satisfying one of the boundary conditions.
other wise the log function is unbounded at zero and
to give zero at 1.
So the Green's function has the form
Now if we take the Wronskian we get
Now we take the function off of the derivative of y in the self adjoint form and multiply it by the Wronskian and set it equal to negative 1 to get
So we can pick any real numbers for A and B that satisfy this equation so let
So the Green's function is
So the solution to the ODE is
Notice this satisfies the boundary conditions and that
and
If you plug this back into the ODE you get
Thanks a lot TheEmptySet. Let me tell you what I have done. The problem was actually the following equation
with and ,
whose self-adjoint form is
The functions and are respective solutions of the homogeneous equations
with
and
with .
Then, , which yields
............ .
Thus the solution is given by
.
Letting here (assuming that it is possible), I get
..... , which is exactly the same expression you have obtained. But the problem here is that in my case, the Wronskian tends to . What is the reason that the method I have applied above does not work (by considering the homogeneous equation with different boundary conditions separately)?
Thanks.
bkarpuz