> You can code a Python script without knowing how does a operating system work. You cannot understand nor use a theorem without knowing the definitions involved.
Better analogy:
You cannot code a Python script without knowing how to call the APIs you need to implement the functionality you want.
Knowing how an operating operating system works is more like knowing how foundational mathematics is axiomatized. Interesting and occasionally useful, but not immediately necessary for the theory of differential equations.
I think you might even be able to deal with differential equations without involving limits, by simply asserting all necessary theorems about derivatives as axioms. This corresponds to relying on a robust library (= the theorems other mathematicians have proved), without worrying about implementation details (= the proofs of those theorems, which might be much more difficult to understand than the results can be used).
You might be able to teach someone how to solve differential equations mechanically the same way you can teach someone how to press keys on the keyboard to code a Python script. Yes, it works, but it's specific and the moment something is slightly different it will not work. For someone to be able to work with DEs they need to know how to work with derivatives and integration, and I think it will be worse to not introduce the formal definition of a derivative. Giving just the rules of calculation will appear as magic or arbitrary, and even worse, it may lead people to wrong conclusions (such as operating differentials as if they were fractions because that worked with ODEs). And that's not even entering into all the theory of differential equations, which usually in engineering it does not get much attention (existence and unicity is not that worrying for an engineer, you tend to already know that a solution exists as you did an experiment/saw something/etc).
Better analogy:
You cannot code a Python script without knowing how to call the APIs you need to implement the functionality you want.
Knowing how an operating operating system works is more like knowing how foundational mathematics is axiomatized. Interesting and occasionally useful, but not immediately necessary for the theory of differential equations.
I think you might even be able to deal with differential equations without involving limits, by simply asserting all necessary theorems about derivatives as axioms. This corresponds to relying on a robust library (= the theorems other mathematicians have proved), without worrying about implementation details (= the proofs of those theorems, which might be much more difficult to understand than the results can be used).