# The Ordinary Linear Differential Equation in the Space of Distributions

Chapter

## Abstract

In the space of distributions, the derivative must be replaced by the distribution-derivative and, consequently, also differential equations by “distribution-derivative equations.” In the latter, the given and the sought quantities are distributions. To emphasize the analogy to the case of functions, we shall employ here for the designation of distributions lower case letters like .

*f*,*y*, . . . (which are usually reserved for functions) instead of the letters*T*,*U*, ⋯. A distribution-derivative equation with constant coefficients has the form:$${D^n}y + {c_{n - 1}}{D^{n - 1}}y + \cdots + {c_1}{D_y} + {c_0}y = f$$

(1)

## Keywords

Weighting Function Impulse Response Integrable Function Classical Function Ordinary Linear Differential Equation
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© Springer-Verlag Berlin Heidelberg 1974