Mother's name on child's birth certificate different than her indian passport

[Hardik]

On our child's birth certificate we have got mother's birth (maiden) name but in the US, all her documents including driving license, Indian passport, social security everything has her married name. 

Will there be any problems since mother's name on child's all the documents do not match with her documents? Should we go ahead and get the name change processed on the birth certificate? 

Need help on this as soon as possible because we want to apply for our kid's passport for our upcoming travel out of the country. Thank you so much!