The integral equation is solved as it is in the case of brownian motion and brownian bridge. The eigenfunctions are sine functions, and the tricky parts are the eigenvalues and the distribution of the random coefficients in the K-L expansion.
If g is the eigenfunction with eigenvalue \gamma, then \gammag = -lambdag and g(0) = 0. If you substitute back into the integral equation for g then you get an equation for \gamma in terms of sine, cosine, and \lambda.
I have recently submitted a manuscript for publication giving the complete solution for the eigenfunction/eigenvalue part of this problem and some generalizations.
Prof Eric Key Dept of Math Sci UW-Milwaukee.