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
If you plug this back into the ODE you get