In a young baby the most common cause of a cough as you descibe here is reflux when your baby breathes in tiny amounts of his feed. Your baby does not appesr to have a cold and he does not appear to have a chest infection. Also the Betanoid which is a very powerful cortisone did not help the cough. The chances of reflux being the cause is high. Please go back to your paediatrician and tell him to test your baby for reflux and to treat your baby if he has this condition. He must not get any more of the Betanoid.
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