A Uniqueness Result on Ordinary Differential Equations with Singular Coefficients

We consider the uniqueness of solutions of ordinary differential equations where the coefficients may have singularities. We derive upper bounds on the order of singularities of the coefficients and provide examples to illustrate the results. 1. Results and examples Classical results on the existence and uniqueness of ordinary differential equations are mostly concerned with continuous coefficients (ref. [1]). Here we consider the uniqueness of ordinary differential equation solutions of coefficients with singularities. We study upper bounds on the order of singularities of the coefficients that guarantee the uniqueness of the solution. Main theorems are stated below. Two examples are given to illustrate and to address the sharpness aspect of the results. Proofs are provided in the subsequent section. Theorem 1. Let f(x) ∈ C∞(−a, a) be a solution (real or complex) of (1) y + an−1(x, y)y(n−1) + · · ·+ a0(x, y)y = 0, x ∈ (−a, a), a > 0 with initial conditions f(0) = f ′(0) = · · · = f (n−1)(0) = 0. If (2) lim x→0 |x|n−k |ak(x, y)| ≤ 1 e , k = 0, 1, · · · , n− 1, where e is the Euler’s number, then there exists δ > 0 such that f ≡ 0 on [−δ, δ]. Remarks: For fixed n, the inequality (2) can be relaxed to (3) lim x→0 |x|n−k |ak(x, y)| < 1 Bn , k = 0, 1, · · · , n− 1, Bn = n−1 ∑