A Simple Proof of the Binomial Theorem Using Differential Calculus

Pythagoras, Binomial, and de Moivre Revisited Through Differential Equations

“…By uniqueness, y 1 (t) = y 2 (t) for all 0 < t < 2π, which on comparing real and imaginary parts recover the two formulas in (14). The present approach highlights the importance of the uniqueness theorem of initial value problems of linear ordinary differential equations in obtaining classical results such as the present ones.…”

“…106-107], [11] using mathematical induction are also well known. In fact several other proofs of the binomial theorem can be found in the literature using combinatorial [12], probabilistic [13], and calculus approaches [14,15]. However, the proofs using ordinary differential equations are not known much.…”

105.42 Generalised binomial theorem via Laplace transform technique

“…Apart from the inductive proof there are several other proofs available for the binomial theorem. One can refer to Fulton [4] for a combinatorial proof, Rosalsky [5] for a probabilistic proof, Hwang [6] and Kataria [7] for proofs based on the theory of differential calculus, and Kataria [8] for a proof based on the Laplace transform technique. In contrast to the binomial theorem, only a few proofs are available for the GBT.…”