IMO, most undergrads taking an undergrad DifEq course will have little understanding of what they are actually doing and just treat it as symbol manipulation. I have seen people encounter the same problem with calculus, matrices, or even simple algebra. And you can keep pushing, but at some point if you want to progress you need to revisit older topics and hopefully gain new understanding.
It's like revisiting multiplication and division in binary and suddenly understanding why a similar process in base 10 works.
PS: Pure symbol manipulation is still powerful, but intuition takes a little more.
But, there is a huge gap between being able to pass a test on X after taking a class and being ok when someone writes the same idea in yet another notation. Or worse reading something when they go step 20 -> step 21 and yes you can get from A to B with an hour of concerted effort and step 20 -> step 21 and see what they mean and could get there the long way in a few minutes of calculation.
Now personally, I only got just enough understanding of Math to read a QM book and recognize how much work it would be to get more than a vague I kind of see what your doing thing going. But, sure if you mean the starting point for monumental effort then yea a solid undergrad math background is 'enough'.