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Non-Existence of Finite Order Solution of Non-homogeneous Second Order Linear Differential Equations
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Abstract
In this paper, we have considered second order non-homogeneous linear differential equations having entire coefficients. We have established conditions ensuring non-existence of finite order solution of such type of differential equations.
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We show that all non-trivial solutions of complex differential equation are of infinite order if coefficients A(z) and B(z) are of special type and establish a relation between the hyper-order of these solutions and the orders of coefficients A(z) and B(z). We have also extended these results to higher order complex differential equations.
We consider the equation
with linearly independent solutions f1,2, where A(z) is a transcendental entire function of finite order. Conditions are given on A(z) which ensure that max{λ(f1),λ(f2)} = ∞, where λ(g) denotes the exponent of convergence of the zeros of g. We show as a special case of a further result that if P(z) is a non-constant, real, even polynomial with positive leading coefficient then every non-trivial solution of
satisfies λ(f) = ∞. Finally we consider the particular equation
where K is a constant, which is of interest in that, depending on K, either every solution has λ(f) = ∞ or there exist two independent solutions f1, f2 each with λ(fi) ≤ 1.
This paper deals with the differential equation , where A and B are analytic on an angle. Some results concerning the estimates of the growth of solutions of the equation restricted in the angle are obtained. By using these estimates we also improve some known results in the plane case. AMS No. 34A20, 30D35
This paper is devoted to studying the growth of solutions of equations of type f″+h(z)eazf′+Q(z)f=H(z) where h(z), Q(z) and H(z) are entire functions of order at most one. We prove four theorems of such type, improving previous results due to Gundersen and Chen.
On Solutions of Second Order Complex Differential Equations, accepted in Bull
- D Kumar
- S Kumar
- M Saini
D. Kumar, S. Kumar and M. Saini, On Solutions of Second Order Complex
Differential Equations, accepted in Bull. Calcutta Math. Soc. 1-9.
290-300. department of mathematics, deen dayal upadhyaya college, university of delhi, new delhi-110078
- C Zongxuan
C. Zongxuan, The growth of solutions of f ′′ + e −z f ′ + Q(z)f = 0 where the
order (Q)=1, Sci. China Math. 45 (2002), no. 3, 290-300.
department of mathematics, deen dayal upadhyaya college, university of delhi, new delhi-110078, india.
E-mail address: dinukumar680@gmail.com