We consider the stability analysis of switched systems under arbitrary switching laws. A powerful approach for addressing this problem is based on studying the "most unstable" switching law (MUSL). If the switched system is stable for the MUSL, it is stable for any switching law. The MUSL can be defined as the solution to a certain optimal control problem. The main analysis tool is then the celebrated maximum principle (MP). This provides an implicit characterization of the MUSL in terms of the adjoint and switching function defined in the MP. In this paper, we consider the optimal control problem defining the MUSL for the particular case of positive linear switched systems. We show that in this case the MP can be strengthened. Specifically, there exist regions in the state-space for which the MUSL is known explicitly.