I'm not sure what automatic differentiation has to do with solving diff. equations but...
I assume that all those automatic methods you mention are passed down from above in some sort of holy scriptures that we're supposed to blindly believe and use? Or maybe some "pencil pushing" mathematician came up with them first and _proved_ that they actually work?
If you're solving DE's by hand you were probably just in a class taught by members of the Mathematician-Teaching Complex (allusion to Military-Industrial Complex).
I assume that all those automatic methods you mention are passed down from above in some sort of holy scriptures that we're supposed to blindly believe and use? Or maybe some "pencil pushing" mathematician came up with them first and _proved_ that they actually work?