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To check the first one

sage: bool(diff(F1) == F2^2)
True

However, I doubt the others are correct as evaluating at a specific x the difference you do not get a zero

sage: B = diff(F2) - (F1*F2 + G1)
sage: B.subs(x=0)
1
sage: C = diff(G1) - (F2*(F2-G2)+F2)
sage: C.subs(x=1).n()
10.5654556363032
sage: D = diff(G2) - (F2*(F1-G1)+G1)
sage: D.subs(x=0)
1