It would be likely Allergic, so following measures will be helpful;
1. T Montek-Lc one tab HS x 3 days
2.syp codistar. 10 ml twice a day x 3 days, if dry persisting cough
3.steam inhalation
Next Steps
If this is not helpful for the coming days, we would go with blood routines and start on antibiotics
Health Tips
Avoid dust pollen animal dander