Self-Adjoint Differential Equation

a differential equation that has the same solutions as its adjoint equation. An ordinary self-adjoint differential equation of even order 2m has the form

and of odd order 2m—1 has the form

where the A_i are functions of x. The concept of self-adjoint differential equations plays an important role in the theory of ordinary and partial differential equations. Under certain boundary conditions, the lefthand part of a self-adjoint differential equation defines a self-adjoint differential operator. Second-order self-adjoint differential equations are the most important in applications.