It's quite hard to know Bro, but I think the next:
حث: to urge/encourage someone "continously" to do something and mostly that happens while that person is doing that work, no matter from time to time or in that moment
يحرِّض : also to encourage /egg on to do something, but this verb can never be used while the person "you mean" is doing that thing right in that moment, maybe from time to time but not quite during your encouraging, also in this case the person who is urging is not doing that so intensively like the first verb.
That's what I could help and excuse me that I can't give for examples because they won't help you much.
You might meet a verb called " يحضّ" and it is pretty meaning the same of "يحث"... Quite no difference.
Anyway, you're free to choose one of the three former verbs up. And as long as you're not talking to a high high professor in Arabic literature, everything is fine:))
Enjoy your time, wish you much fun..