I do try to make my son do his homework, but he refuses to cooperate.
I did think you were going to speak to him about it.
I do hope he’ll try harder this year.
In each example, do is used to add emphasis to the main verb, that is, to make the expression or feeling stronger. In these examples, do functions as an emphatic auxiliary. And I just want to mention here that the auxiliary do cannot be combined with any other auxiliary – that is, we can’t say
I must do try to make my son do his homework.
although we could say
I must try to make my son do his homework.
You mention do used with believe, Hossein; can you see how it adds emphasis in the following examples?
I do believe we’ve met somewhere before.
I do think Chinese is a difficult language to learn.
I do feel that Jordan’s is the best restaurant in town.
I do hope she’ll be happier in the new house.
In all of these examples, do is used to reinforce the strength of claim and show certainty. But there’s another usage of do as an emphatic auxiliary. Sometimes we can use do to contradict or show contrast and here are some more examples:
I did call on him yesterday morning, although he said he didn’t hear the doorbell.
The office staff said my fax was late, but I did send it on time.
My daughter is so naughty at home, but her teacher says she does study hard at school.
In each of these examples, do shows the contrast between the expected and real outcome in each situation, and in speech, an emphatic do would usually be stressed. Well, Hossein, I do hope that this explanation has been useful to you!