Acute Viral Bronchitis: This is the most common cause of acute bronchitis, especially in the elderly. The presentation of cough and expectoration for 10 days in a 75-year-old female is consistent with a viral etiology, which often resolves on its own but can be prolonged in older adults due to decreased immune response and possible underlying chronic conditions.

Acute Bacterial Bronchitis: While less common than viral causes, bacterial bronchitis can occur, especially if there's a secondary bacterial infection following a viral illness. The elderly are more susceptible to bacterial infections due to age-related immune system decline.
Influenza: Given the seasonal nature of influenza and its potential to cause severe illness in the elderly, it should be considered, especially if the symptoms started during influenza season.
Chronic Obstructive Pulmonary Disease (COPD) Exacerbation: If the patient has a history of COPD, an exacerbation could present with increased cough and expectoration. The elderly are more likely to have COPD due to years of potential exposure to irritants like cigarette smoke.

Pneumonia: Although the patient's symptoms are suggestive of bronchitis, pneumonia can present similarly, especially in the elderly, who may not exhibit the typical symptoms of fever, chills, and high fever. Pneumonia is a critical diagnosis to consider due to its potential for severe morbidity and mortality in this age group.
Pulmonary Embolism: This condition can sometimes present with cough and may be considered, especially if there are other risk factors such as recent immobilization, cancer, or history of deep vein thrombosis.
Lung Cancer: In an elderly patient with a new onset of cough and expectoration, especially if there's a history of smoking, lung cancer should be considered, although it's less likely to present acutely.

Tuberculosis (TB): While less common in many parts of the world, TB can present with chronic cough and should be considered, especially if the patient has risk factors such as immunocompromised status, recent travel to or from areas with high TB prevalence, or exposure to someone with TB.
Cystic Fibrosis: Extremely rare to be diagnosed for the first time in a 75-year-old, but could be considered if there's a history suggestive of recurrent respiratory infections or malabsorption.
Bronchiectasis: This condition involves permanent enlargement of parts of the airways and can cause chronic cough and expectoration. It's less likely to be diagnosed for the first time in an elderly patient without a history of recurrent infections or other suggestive symptoms.