'I give the medicine to my child' is the correct option; to me 'feed' suggests some form of nourishment, rather than medical treatment, and generally would not be used in this particular context. 'To feed' is actually more commonplace when talking of animals, for example: "I fed the cat" Or: "The horses were feeding on the hay" The only real instance where it could be used to describe humans is if you are comparing them to animals in some way.
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In this case, let s suppose the child has a disability, for example and so he/she cannot take the medicine by him/herself. We have to put medicine in his/her mouth, give him water, and even encourage him swallow the medicine. So, in such situation, which verb should be used? 'give' does not seem to be perfect for this case. Any idea?