Stay the course跟Persist in the end 之間的用法上有什麼差別？

1. If you "stay the course." you finish something that you have started, even though it has become very difficult.
The oldest president in American history had stayed the course for two terms.
p.347 "course" Collins Cobuild English Dictionary for Advanced Leraners

2. You say "in the end" when you are saying what is the final result of a series of events, or what is your final conclusion after considering all the relevant facts.
I toyed with idea of calling the police, but in the end I didn't
p.507 "end" Collins Cobuild English Dictionary for Advanced Learners

3. If you "persist" in doing something, you continue to do it, even though it is difficult or other people are against it.
Why does Britain persist in running down its defence forces?
see p.1147 "persist" Collins Could English Dictionary for Advanced Learners

persist in ~ing才是正確的用法，你的persist in the end寫法可能不正確，因為缺了後面的動名詞，就語意不全了，介系詞片語in the end只是用來標示時間座標的。

Stay the course==continue going until the end; Persist in the end==continue to exist to the end or occurring again and again to the end. Yip