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If one forgets to say Ya'aleh V'yavo on the eve of rosh chodesh, one does not repeat:

If one remembers immediately after completing the blessing of Retzeh, he should insert Ya’aleh VeYavo there, and then continue with Modim. If one already began Modim, but remembered before moving his feet back at the end of Shemoneh Esrei, he should return to the beginning of Retzeh and continue from there until the end. The preceding is true only regarding Shacharit and Mincha, but if one forgot to say Ya’aleh VeYavo in Ma’ariv, he does not go back, unless he remembers before saying God’s name at the conclusion of Retzeh. The reason Ma’ariv is different is that the Beit Din did not sanctify the new moon at night. Thus, even though one should, ideally, say Ya’aleh VeYavo in Ma’ariv, he does not repeat the Shemoneh Esrei, or even just one blessing, in order to say it (Berachot 29b, 30b; Shulchan Aruch, Orach Chaim 422:1)

From http://ph.yhb.org.il/en/05-01-10/

But what about ma'ariv at the beginning of the second day of rosh chodesh? The day has already been sanctified and established -- we had a full day of it.

  • What if you Daven a weekday Shmone Esrei on Rosh HaShana? – Double AA Nov 2 '16 at 15:12
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The same rule applies both nights (ShA OC 422:1).

(Indeed while this seems to be explicit in the Gemara (Brachot 30b), the Beit Yosef (ibid.) notes with amazement that the Orchot Chayim quotes two rabbis who disagreed and required going back on the second night.)

Mishna Berura (ibid. sk 3) explains that each day of Rosh Chodesh doesn't affect the other as each is just a "doubt" about which is the right day (sefeika deyoma). There's only really one day of Rosh Chodesh: the first of the month.

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