Adjoint Differential Equation

a concept of the theory of differential equations. The adjoint of the differential equation

is the differential equation

The relation is symmetric: L(y) = 0 is also the adjoint of M(z) = 0.

Adjoint differential equations satisfy the identity

The expression ψ(y, z) is known as the bilinear concomitant. It is linear in y and z and in the derivatives of y and z through the (n – 1)st order. If we know k integrals of the adjoint equation, we can decrease the order of the original equation by k.

If

(3) y₁, y₂, ..., y_n

is a fundamental system of solutions of equation (1), then the fundamental system of solutions of equation (2) is given by the formulas

where i = 1, 2, ... , n and Δ is the Wronskian of system (3). If boundary conditions are given for equation (1), then there exist for equation (2) adjoint boundary conditions such that equation (1), equation (2), and the corresponding boundary conditions determine adjoint differential operators.

The concept of adjoint differential equations can be extended to systems of differential equations and to partial differential equations.