W. Magnus' classical paper on the exponential solution of differential equations for a linear operator received much attention since its appearance in 1954, both in applied mathematics as well as physics. More recently, Magnus' results were explored from a more algebraic-combinatorial perspective using the language of operads and combinatorial Hopf algebras. In this talk we review recent work on the classical Magnus expansion, unveiling a new structure by using the language of dendriform and pre-Lie algebras as well as Hopf algebras of rooted trees. Let us emphasise that a dendriform algebra may be seen at the same time as an associative as well as a (pre-)Lie algebra. Main examples of dendriform algebras are provided by the shuffle and quasi-shuffle algebras as well as Rota--Baxter operators, e.g. the Riemann integral map or Jackson's q-integral. We introduce the notion of linear equations in dendriform algebras, which include matrix initial value problems as a particular example, and investigate its solution theory. Due to the intimate link between dendriform algebras and Rota-Baxter operators, motivations as well as applications naturally appear. They rang from numerical methods to the process of renormalisation in perturbative quantum field theory. (This is joint work with D. Manchon).