The birth certificate is issued everywhere without the name of the new-born. So that would not cause any problem. For the purpose of mother's name, the candidate is required to write it afresh in the application form properly. There is generally no need to tally it with the birth certificate. However, if the difference in the name of the mother is pointed out by the school authorities, an advertisement is required to be given in the newspaper stating that 'X' and 'Y' are the same person.