  We study the inverse problem in the difference Galois theory of linear
differential equations over the difference-differential field $\mathbb{C}(x)$
with derivation $\frac{d}{dx}$ and endomorphism $f(x)\mapsto f(x+1)$. Our main
result is that every linear algebraic group, considered as a difference
algebraic group, occurs as the difference Galois group of some linear
differential equation over $\mathbb{C}(x)$.
