For the passage of time are both of these interchangeable?
What nuances does using either over the other have?
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過ごす is clearly different from the other two. Its subject is usually a person (私は東京で1日を過ごした). It never takes a time length as a subject. (× 5か月が過ごす).
経つ and 過ぎる are very similar and usually interchangeable when the subject is a direct time length like 3日, 半年, 50年. In this case, 経つ is more objective, whereas 過ぎる may be slightly more emotional or nostalgic. When the subject is a noun that represents an event or a season, you can only use 過ぎる. For example you can say 夏休みが過ぎる／クリスマスが過ぎる but not 夏休みが経つ／クリスマスが経つ.