Welcome to Justanswer! I am Dr. Altman and happy to assist you both today!
So sorry to hear Archie has this cough!
There are some home remedies that can help reduce the cough and give relief but with his age if there is no relief please consider a veterinary exam to ensure there is no underlying cause for concern such as pneumonia, heart disease, collapsing trachea that requires medical management
1.) The steam of the shower when the cough is severe or constant
2.) A little honey three times daily can help soothe the lungs
3.) If you have a humidifier or nebulizer that would also be wonderful to use in this situation
Unfortunately I prefer to avoid cough suppressants because when suppressing a cough if there is phlegm or buildup and lead to bronchitis or even pneumonia if the material is not coughed up so this is not something I tend to administer. And Robitussin DM is actually not advised in pets because the DM part is toxic to dogs
I hope this information makes sense! If the cough is productive, there is lethargic or a reduced appetite then it would be best to have a veterinary exam to ensure antibiotics are not warranted.
Please let me know if I can be of any further assistance to you both!
If you have more questions or if I can help in any other way, please do not hesitate to ask! If you would like to accept my answer, please press RATE OUR CONVERSATION (I am not compensated in any other way). Bonuses are always welcome. Thanks!