The guy I was responding to, mikekchar, claimed that he was given advanced physics problems requiring diff eqs without having studied them before; this is absolutely not the case in any physics program I've ever heard of. By the time you get to Junior Year physics, you've had diff eqs, or at least an introduction to the ordinary diff eqs you get in late Calculus. You can probably learn them both in the same course; but only if the course is designed for this purpose (aka Boas based "math methods for physicists" or something).
I do remember some guys in grad school who had never seen a Greens function before (I was doing them in sophomore years; guess I was lucky) encountering JD Jackson, but that's about the only disconnect I've seen.
Just speaking from my own expedience, I had advanced (jr year) mechanics before diff eq... but admittedly was taking it early. It was doable but tough.
Greens function were new to me in grad school. Grad school mechanics had be learning math I didn’t know before too... can’t recall what it was at the moment.
And then the advanced AMO class had some more math I hadn’t seen.
I do remember some guys in grad school who had never seen a Greens function before (I was doing them in sophomore years; guess I was lucky) encountering JD Jackson, but that's about the only disconnect I've seen.