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1.
We consider transcendental meromorphic solutions with N(r,f) = S(r,f) of the following type of nonlinear differential equations:f~n + Pn-2(f) = p1(z)e~(α1(z)) +p2(z)e~(α2(z)),where n≥ 2 is an integer, Pn-2(f) is a differential polynomial in f of degree not greater than n-2 with small functions of f as its coefficients, p1(z), p2(z) are nonzero small functions of f, and α1(z), α2(z)are nonconstant entire functions. In particular, we give out the conditions for ensuring the existence of meromorphic solutions and their possible forms of the above equation. Our results extend and improve some known results obtained most recently. 相似文献
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The main purpose of this article is to study the existence theories of global meromorphic solutions for some second-order linear differential equations with meromorphic coefficients, which perfect the solution theory of such equations. 相似文献
3.
On a certain type of nonlinear differential equations admitting transcendental meromorphic solutions
We study the differential equations w 2+R(z)(w (k))2 = Q(z), where R(z),Q(z) are nonzero rational functions. We prove
- if the differential equation w 2+R(z)(w′)2 = Q(z), where R(z), Q(z) are nonzero rational functions, admits a transcendental meromorphic solution f, then Q ≡ C (constant), the multiplicities of the zeros of R(z) are no greater than 2 and f(z) = √C cos α(z), where α(z) is a primitive of $\tfrac{1} {{\sqrt {R(z)} }}$ such that √C cos α(z) is a transcendental meromorphic function.
- if the differential equation w 2 + R(z)(w (k))2 = Q(z), where k ? 2 is an integer and R,Q are nonzero rational functions, admits a transcendental meromorphic solution f, then k is an odd integer, Q ≡ C (constant), R(z) ≡ A (constant) and f(z) = √C cos (az + b), where $a^{2k} = \tfrac{1} {A}$ .
4.
Octavian G. Mustafa 《Journal of Mathematical Analysis and Applications》2004,294(2):548-559
We are concerned with the nonexistence of L2-solutions of a nonlinear differential equation x″=a(t)x+f(t,x). By applying technique similar to that exploited by Hallam [SIAM J. Appl. Math. 19 (1970) 430-439] for the study of asymptotic behavior of solutions of this equation, we establish nonexistence of solutions from the class L2(t0,∞) under milder conditions on the function a(t) which, as the examples show, can be even square integrable. Therefore, the equation under consideration can be classified as of limit-point type at infinity in the sense of the definition introduced by Graef and Spikes [Nonlinear Anal. 7 (1983) 851-871]. We compare our results to those reported in the literature and show how they can be extended to third order nonlinear differential equations. 相似文献
5.
Maria V. DeminaNikolai A. Kudryashov 《Applied mathematics and computation》2011,217(23):9849-9853
The problem of constructing and classifying exact elliptic solutions of autonomous nonlinear ordinary differential equations is studied. An algorithm for finding elliptic solutions in explicit form is presented. 相似文献
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By utilizing Nevanlinna's value distribution theory of meromorphic functions, it is shown that the following type of nonlinear differential equations:
fn(z)+Pn−3(f)=p1eα1z+p2eα2z