On the Dirichlet problem for ordinary differential equations

We consider a fourth order differential operator Ly = (ry")"-(qy')'+py and a sixth order differential operator My= -(mym )"+(ry")"-(qy')'+py on [a,∞) where m", r", q' and p are continuous, m is positive, r and q are nonnegative and p is bounded below by a positive constant.] The Dirichlet index of L and M is the dimension of the solution set to My = O and Ly = O satisfying

a r(y")2 q(y')2 py ds

and ${r(y)²2 +q(^)²} + p^{2}ds

respectively.

We put conditions on the coefficients of L and M to insure that the Dirichlet index is two and three respectively. Applications of the theorems are used to show that this occurs when the coefficients are Q1(t)exp[Q2(t)| where Q1 and Q_2 are finite sums of real multiples of real powers of t.