I have always found integration to be more intuitive than derivation. That makes me wonder why we see so many differential equations in applied mathematics...
Looking through Maxwell's equations, I see that all of them can be formulated both in integral and differential form. Is that true for all differential equations? For instance, can Schrodinger's equation be written in integral form?
It doesn't look like it. Would the path integral formulation work for your question about Schrodinger's equation?