B[u_] = ρ[1]/ρ[u]

e1 = 
      Numerator[Together[B''[1] + B'[1] - B'[1]^2]]

e2 = u h[ρ[u]] - Subscript[ρ, g]

e3 = D[e2, u] /. u -> 1

e4 = D[e2, u, u] /. u -> 1

e5 = {e1, e3, 
          e4} /. {ρ''[1] -> ρ2, ρ'[
              1] -> ρ1, ρ[1] -> ρ0, 
          h[ρ[1]] -> h0, h'[ρ[1]] -> 
            h1, h''[ρ[1]] -> h2}

GroebnerBasis[e5, {ρ2, ρ1, ρ0, h0, h1, h2}]

Eliminate[e5 == 0, {ρ2, ρ1}]