Missing coefficient for Series Solution for DE

[dietcookie]

I can't for the life of me figure out where C$_{0}$ went. I scanned my work here..I've looked back and forth through my book and other texts, it always seems like all the coefficients are accounted for and/or they equal zero. As it stands, I only have C$_{1}$ Thanks!

Full size http://i.imgur.com/Cuczu.jpg

 

[haruspex]

I agree that 2C₂+C₀ = 0, but by what logic did you deduce they're both 0?