  Let $K$ be a field of characteristic zero, let $\sigma$ be an automorphism of
$K$ and let $\delta$ be a $\sigma$-derivation of $K$. We show that the division
ring $D=K(x;\sigma,\delta)$ either has the property that every finitely
generated subring satisfies a polynomial identity or $D$ contains a free
algebra on two generators over its center. In the case when $K$ is finitely
generated over $k$ we then see that for $\sigma$ a $k$-algebra automorphism of
$K$ and $\delta$ a $k$-linear derivation of $K$, $K(x;\sigma)$ having a free
subalgebra on two generators is equivalent to $\sigma$ having infinite order,
and $K(x;\delta)$ having a free subalgebra is equivalent to $\delta$ being
nonzero. As an application, we show that if $D$ is a division ring with center
$k$ of characteristic zero and $D^*$ contains a solvable subgroup that is not
locally abelian-by-finite, then $D$ contains a free $k$-algebra on two
generators. Moreover, if we assume that $k$ is uncountable, without any
restrictions on the characteristic of $k$, then $D$ contains the $k$-group
algebra of the free group of rank two.
