Is it permissible to fast the second day of Ramadan without renewing the intention, if the intention to fast was present from the beginning of the month but the first day was not fasted due to a legitimate excuse?

If a Muslim's heart forms the intention to fast on the night preceding the second day of Ramadan, then his fast is valid. However, if the intention does not cross his mind and he does not resolve to fast, then his fast is not valid, even if he had intended before the first day. This is because the valid time for intention is during the night. An intention made before the night of fasting is not considered.