EIGENVALUE ESTIMATES FOR A GENERALIZED STURM-LIOUVILLE PROBLEM.

A method is presented for approximating the eigenvalues of the general Sturm-Liouville equation left bracket -r(x)y prime right bracket prime plus q(x)y equals lambda p(x)y over the interval left bracket 0,1 right bracket , subject to the boundary conditions y(0) equals y(1) equals 0, where q(x), r(x), and p(x) are non-negative functions with bounded second, third, and fourth derivatives, respectively, on left bracket 0,1 right bracket . The eigenvalue estimates are obtained by dividing the unit interval into m equal subintervals of length h and solving the differential equation on each subinterval by making linear, quadratic, and cubic approximations to q(x), r(x), and p(x), respectively. It is shown by means of a theorem, and illustrated by several examples, that the eigenvalue estimates obtained by this method differ from the exact eigenvalues by a quantity of order b**2.