  In this paper an application of the recently introduced pre-Lie Magnus
expansion to Jackson's q-integral and q-exponentials is presented. Twisted
dendriform algebras, which are the natural algebraic framework for Jackson's
q-analogues, are introduced for that purpose. It is shown how the pre-Lie
Magnus expansion is used to solve linear q-differential equations. We also
briefly outline the theory of linear equations in twisted dendriform algebras.
