findSeriesSolution[y_,x_,nTerms_]:=Module[
{roots,pt=0,ode,s0,s1,ic,eq,sol},ic={y[0]->0,y'[0]->0};
  ode=y''[x]-4 x-y[x]^2;
  s0=Series[ode,{x,pt,nTerms}];
  s0=s0/.ic;
  roots=Solve@LogicalExpand[s0==0];
  s1=Series[y[x],{x,pt,nTerms+2}];
  sol=Normal[s1]/.ic/.roots[[1]]
]

findSeriesSolution[y,x,5]
(*  (2 x^3)/3 *)