y = a[0] + a[1]*x + Sum[ a[i]*x^i,{i,2,20}] +O[x]^21

eqn = D[y,{x,2}] - y == 0

expr = LogicalExpand[eqn]

list = Table[a[i],{i,2,20}]

sol = Solve[expr, list]

norm = Normal[y] /. sol[[1]]

y1[x_] = Coefficient[norm, a[0]]

Series[Cosh[x],{x,0,20}]

y2[x_] = Coefficient[norm, a[1]]

Series[Sinh[x],{x,0,20}]