Elfen Lied: Just How Did the Young Girl Arrive at Those Feelings?(2005)

This episode takes place within episode 11 as Nana attempts to fit in the household while trying to find a chore that she can do and get used to Nyu's presence whom she only knows as Lucy. Running away after getting scolded, Nana ends up in the beach and meets Bando whom she promised to bring Lucy to later that night. Leaving the household with Nyu, it begins to rain and Nyu slips, hits her head, and is knocked unconscious. As Bando waits and everyone else worries where they might be, Nana, with an uncoscious Nyu, takes shelter as Lucy's consciousness begins to come out and dreams an old memory of how she was captured and why she doesn't attack Director Kurama. As the rain begins to let up, Kouta and the others find Nana and Lucy and takes them home. There, Nana finally finds the chore that she will be doing - feeding the dog.

(Source: Anime News Network)