Concept:
Dry cough is non-productive cough, meaning mucus is not produced.
Step 1: Dry cough requires a cough suppressant.
Step 2: Dextromethorphan is an antitussive agent.
Step 3: It suppresses cough reflex and is suitable for dry cough.
Step 4: Antibiotics are not needed without evidence of bacterial infection.
\[
\boxed{\text{Dextromethorphan}}
\]