2) is correct.

3) Let n=N-4 and rewrite the last sum, starting from N=4. After, the character "N" can be replaced by the character "n".

For information : The EDO is one of Bessel kind. Solutions are :

y = C1 x^(3/2) BesselJ[1/2 , x^3] +C2 x^(3/2) BesselJ[-1/2 , x^3]