# Self-Adjoint Differential Equation

## 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.