  In this study, linear second-order conformable differential equations using a
proportional derivative are shown to be formally self-adjoint equations with
respect to a certain inner product and the associated self-adjoint boundary
conditions. Defining a Wronskian, we establish a Lagrange identity and Abel's
formula. Several reduction-of-order theorems are given. Solutions of the
conformable second-order self-adjoint equation are then shown to be related to
corresponding solutions of a first-order Riccati equation and a related
quadratic functional and a conformable Picone identity. The first part of the
study is concluded with a comprehensive roundabout theorem relating key
equivalences among all these results. Subsequently, we establish a Lyapunov
inequality, factorizations of the second-order equation, and conclude with a
section on boundary value problems and Green's functions.
