Applications to Ordinary Differential Equations

Abstract

In Section 2.6 we defined the differential operator L,

$$ \begin{array}{*{20}c} {Lt = \left( {a_n (x)\frac{{d^n }} {{dx^n }} + a_{n - 1} \frac{{d^{n - 1} }} {{dx^{n - 1} }} + \cdot \cdot \cdot + a_1 \frac{d} {{dx}} + a_0 } \right)t} \\ { = \sum\limits_{m = 0}^n {a_m (x)\frac{{d^m }} {{dx^m }},} } \\ \end{array} $$

(1)

and its formal adjoint L*,

$$ L*\varphi = \sum\limits_{m = 0}^n {( - 1)^m d^m (a_m (x)\varphi )/dx^m ,} $$

(2)

where the coefficients a _m (x) are infinitely differentiable functions, t is a distribution, and ø is a test function. These operators are related by the equation

$$ \left\langle {Lt,\varphi } \right\rangle = \left\langle {t,L*\varphi } \right\rangle . $$

(3)