Who wrote the piano piece in "House" The Half Witt?

Answer 1

After several edits, this is my final answer (from the two sources listed below).
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The piano music House and the Patient of the Week were playing together:
1) Boomtown Rats' "I Don't Like Mondays."
2) The song that was supposed to have been written by House while he was in junior high was written specifically for this episode by the people responsible for the music on the show.
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The first song that House plays on the piano and Patrick repeats is "I Don't Like Mondays" by the Boomtown Rats. The classical piece Patrick performs on stage is Beethoven's Waldstein Sonata (No. 21 in C major, Op. 53, Allegro con brio). The song playing when Patrick is having the internal EEG is "Rainy Day Lament" by Joe Purdy. The song playing during the closing scenes is "See The World" by Gomez.