Is it permissible to set a specific day for the birth of a baby using artificial induction after exceeding the expected due date?

There is nothing wrong with a doctor specifying a day for delivery, because it is based on an approximate calculation, not a claim to knowledge of the unseen. It is permissible to use induced labor for a benefit, as long as it does not harm the mother or the fetus, especially if natural labor is delayed. Ibn Uthaymeen mentioned the permissibility of taking lawful medications that help a woman in childbirth if it becomes difficult for her.