It really depends on what kind of modeling you want to do. Like it's been said many times above, to achieve more physical realism you'll have to resort to ODEs sometime. I think taking a course on ODEs before numerical methods would probably be useful. Numerical methods are what you're going to need if you ever want to actually apply your ODE knowledge, but understanding what it is that the numerical methods are actually doing is important.
and ODEs like the heat and wave equations are themselves solved by eigendecomposing a linear operator (the sines/cosines are its eigenvectors).
Not to be too nitpicky, but as far as I remember, the heat and wave equations are second-order PDEs, not ODEs.