Is using the explanations and solutions for homework assignments provided by an online private tutor, after obtaining the doctor's permission to refer to the problems for confirmation or to learn the method of solution, considered unlawful (haram)?

The reference for seeking help from others in completing assignments is what the course instructor determines. What you mentioned about the instructor allowing you to refer to the problem to verify the correctness of the solution, or to refer to the solved textbook to understand the steps and then solve another example yourself, does not include his permission for someone else to complete your assignments. Therefore, it is not permissible for you to do so, because it is cheating. The Prophet, peace and blessings be upon him, said: "Whoever cheats is not one of us."