We will consider coalgebras of the type $F(X) = mathbb{R} imes X$, representing streams over real numbers. For such streams, we coinductively define a number of operators (including, but not limited to, addition and multiplication), and we furthermore consider coinductive proofs through bisimulation. We will discuss polynomial and rational streams, and give a characterization of the latter through so-called 'behavioural differential equations' of a specific type. If time allows, we will briefly discuss the relation between the stream calculus and formal languages.