Phylogenetic diversity (PD) is a useful metric for selecting taxa in a range of biological applications, for example, bioconservation and genomics, where the selection is usually constrained by the limited availability of resources. We formalize taxon selection as a conceptually simple optimization problem, aiming to maximize PD subject to resource constraints. This allows us to take into account the different amounts of resources required by the different taxa. Although this is a computationally difficult problem, we present a dynamic programming algorithm that solves it in pseudo-polynomial time. Our algorithm can also solve many instances of the Noah's Ark Problem, a more realistic formulation of taxon selection for biodiversity conservation that allows for taxon-specific extinction risks. These instances extend the set of problems for which solutions are available beyond previously known greedy-tractable cases. Finally, we discuss the relevance of our results to real-life scenarios.