What is procedure to change name of mother in birth Certificate ?

Hello ! I am a 28 years old male. My father's second marriage is not registered. I am my father's son from his second marriage. But somehow my birth certificate does not hold my father's second wife's name, who is my actual biological mother. Instead it holds my father's first wife's name who is not my actual biological mother. Now I want to apply for passport hence need a birth certificate with the true information. What should I do ? Can my birth certificate be corrected ? Can maternal DNA test help me in anyway ? Thank you