How to find eigenfunctions and eigenvalues of a fourth order ODE?

Dear all,

I am having trouble with the following problem. Suppose we have the fourth order ODE with boundary conditions:

http://r.789695.n4.nabble.com/file/n4591748/problem.jpg

where q(t) is a known function.

Note here the lambda parameter is changing, so essentially we have a series of ODEs. lambda is called an eigenvalue, the solution y is called an eigenfunction associated with lambda. The question is to find all the eigenvalues lambda and eigenfunctions y. Approximate solutions are OK.

I have found a function called bvp4c(), which, however, requires one to give an initial guess of the lambda. I am wondering if there is a more automatic way so that by plugging the coefficient function q(t) we directly get the eigenfunctions and eigenvalues.

Best regards, Jeff