I would like to stress what is also mentioned in a sibling comment: quantum mechanics' original formulation was not in terms of a wave function and differential equations.

Heisenberg's matrix mechanics was the first, and formulated entirely in terms of linear algebra. See for example the introduction of basic linear algebra techniques in the famous Bohr-Jordan paper from 1925:

http://www.psiquadrat.de/downloads/bornjordan1925.pdf