On distribution of zeros of solutions of a second order linear differential equation with polynomial coefficients
نویسندگان
چکیده
منابع مشابه
Sums of Zeros of Solutions to Second Order Differential Equations with Polynomial Coefficients
We consider the equation u′′ = P (z)u, where P (z) is a polynomial. Let zk(u), k = 1, 2, . . . be the zeros of a solution u(z) to that equation. Inequalities for the sums ∑j k=1 1 |zk(u)| (j = 1, 2, . . .) are derived. They considerably improve the previous result of the author. Some applications of the obtained bounds are also discussed. An illustrative example is presented. It shows that the ...
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We consider the equation y′′ = F (z)y (z ∈ C) with an entire function F satisfying the condition |F (z)| ≤ A exp ` |z|ρ ρ ́ (ρ ≥ 1, A = const > 0). Let zk(y), k = 1, 2, . . . be the zeros of a solution y(z) to the above equation. Bounds for the sums j X k=1 1 |zk(y)| (j = 1, 2, . . . ) are established. Some applications of these bounds are also considered.
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We consider the differential equations y = λ0(x)y ′ + s0(x)y, where λ0(x), s0(x) are C −functions. We prove (i) if the differential equation, has a polynomial solution of degree n > 0, then δn = λnsn−1 − λn−1sn = 0, where λn = λ ′ n−1 + sn−1 + λ0λn−1 and sn = s ′ n−1 + s0λk−1, n = 1, 2, . . . . Conversely (ii) if λnλn−1 6= 0 and δn = 0, then the differential equation has a polynomial solution o...
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ژورنال
عنوان ژورنال: Journal of Mathematical Analysis and Applications
سال: 1973
ISSN: 0022-247X
DOI: 10.1016/0022-247x(73)90259-x