On zeros of solution of self-adjoint differential equations.

This study will deal with the zeros of solutions of self-adjoint linear differential equations of second order. In the following we consider some def1n1t1ons and theorems that relate to ordinary differential equations. Definition 1.1. A homogeneous linear differential equation of order n has the form a0y{n ) + a1y(n-l) + ••• +any= 0 where a0 �� 0 and e��ch a1 = a 1(x) is continuous on an interval (a,b), 1 = 1,2, ••• ,n. Definition 1.2. If L(y) 1s a linear operator and L(y) = a0(x)y\u27\u27(x) + a1(x)y\u27(x) + a2 (x)y(x}, then 1ts adjoint, L(z}, 1s denoted by [��0(x)z(x)] \u27\u27 - [a1 (x)z(x)]\u27 + a2(x)z(x). If L(y) = L(y), then the differential equation, L(y): 0 is self-adjoint of second order. number of theorems, some of them included without proof, were used as a basis for the study 1n this thesis. Theorem 1.1. L(y) = 0 is self-adjoint 1f and only if a1(x) = a0•(x). (5, p. 98