  We study two classes of linear difference differential equations analogous to
Euler-Cauchy ordinary differential equations, but in which multiple arguments
are shifted forward or backward by fixed amounts. Special cases of these
equations have arisen in diverse branches of number theory and combinatorics.
They are also of use in linear control theory. Here, we study these equations
in a general setting. Building on previous work going back to de Bruijn, we
show how adjoint equations arise naturally in the problem of uniqueness of
solutions. Exploiting the adjoint relationship in a new way leads to a
significant strengthening of previous uniqueness results. Specifically, we
prove here that the general Euler-Cauchy difference differential equation with
advanced arguments has a unique solution (up to a multiplicative constant) in
the class of functions bounded by an exponential function on the positive real
line. For the closely related class of equations with retarded arguments, we
focus on a corresponding class of solutions, locating and classifying the
points of discontinuity. We also provide an explicit asymptotic expansion at
infinity.
