Could Persistent Coughing And Breathlessness Be Due To Bronchitis?

I am having persistent coughing from 14 days. I feel phelgm but can t spit it out. Feel short of breath. doctor has diagnosed bronchitis but i want to be sure about it. My doctor told me that there is no need for taking X ray, do you think the same ? He gave me a nasal spray, cough syrup, antibiotic and 1 other medicine..i am taking a lot of fluids and i have stopped drinking/eating cold items. Please let me know if the diagnosis done is right ?