What is the name of the background music that plays before Kaori and Kousei's performance? In Episode 4, right before their performance, Kaori headbutts Kousei, and then some BGM starts playing. In the next episode, episode 5, as Kousei is running home in the rain, the same song plays again, and continues into the flashback. What is the name of the song? 

I've transcribed the first part of the song, it gets pretty inaccurate towards the end but should still be recognizable.

http://www.noteflight.com/scores/view/e958304d60b8f1a61108335f228623a5e0084cef
Adding to my previous comment on the question, the official website now has a page for the "Your Lie in April original song &amp; soundtrack" (「四月は君の嘘 ORIGINAL SONG &amp; SOUNDTRACK」). They have a full listing of the soundtrack there, but you cannot listen to it, nor even preview it.
So I did a quick search, and found this Amazon.co.jp page for the cd, where you can actually listen to a preview of every song.

Three songs seem to fit the one you want, given that two of them are only variations on the same theme. The one that does seem to be the exact match is called 友人A君を私の伴奏者に任命します (something in the lines of "I appoint Friend-A-kun as my accompanist"), and is the 18th track on the first cd. You can buy it individually here. The first variation is the 12th track on the first cd, called 今日のことは忘れられないよ (something like "I cannot forget that day"), which, once again, can be bought individually here. And the last one is 7th track on the second cd (or the 40th if you count them continuously, like Amazon does), called 分かりやすいヤツ (something like "Easy-to-understand guy"), which can also be bought individually, here