I've been asked to write an essay on whether the work on PDE's in the 19th century belonged to applied or pure mathematics. Does anyone know of any useful sources I could use?
It belonged to both. For example, the most important work on PDE in 19th century was arguably Fourier's Analytic theory of heat. You don't have to read the book, to conclude that this was applied mathematics, just from the title. Other important work on PDE, comes from pure mathematics (differential geometry for example), or the work of Liouville on the Liouville's equation.
An interesting example is Cauchy-Riemann equations which come from Complex Analysis (pure mathematics) but closely related to fluid dynamics (applied mathematics).
Laplace's equation appeared first in the work on Saturn rings (applied mathematics) but later was used everywhere and its theory (the theory of harmonic functions) was pure mathematics.
In general, the distinction between pure and applied mathematics was never sharp and clear, especially in 19th century. So the question you are assigned has actually little sense.