# Applications to Ordinary Differential Equations

• Ram P. Kanwal

## 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)

## Keywords

Ordinary Differential Equation Classical Solution Fundamental Solution Jump Discontinuity Bessel Equation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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