Oscillation of arbitrary-order derivatives of solutions to the higher order non-homogeneous linear differential equations taking small functions in the unit disc

<abstract><p>In this article, we study the relationship between solutions and their arbitrary-order derivatives of the higher order non-homogeneous linear differential equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f = F(z) \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>in the unit disc $ \bigtriangleup $ with analytic or meromorphic coefficients of finite $ [p, q] $-order. We obtain some oscillation theorems for $ f^{(j)}(z)-\varphi(z) $, where $ f $ is a solution and $ \varphi(z) $ is a small function.</p></abstract>

The J-Generalized P - K Mittag-Leffler Function

We know that the classical Mittag-Leffler function play an important role as solution of fractional order differential and integral equations. We introduce the j-generalized p - k Mittag-Leffler function. We evaluate the second order differential recurrence relation and four different integral representations and introduce a homogeneous linear differential equation whose one of the solution is the j-generalized p-k Mittag-Leffler function. Also we evaluate the certain relations that exist between j-generalized p - k Mittag-Leffler function and Riemann-Liouville fractional integrals and derivatives. We evaluate Mellin-Barnes integral representation of j-generalized p-k Mittag-Le er Function. The relationship of j-generalized p-k Mittag-Leffler Function with Fox H-Function and Wright hypergeometric function is also establish. we obtained its Euler transform, Laplace Transform and Mellin transform. Finally we derive some particular cases.

Linear differential equations with constant coefficients

We consider the second order homogeneous linear differential equation (H) $${ ay'' + by' + cy = 0 }$$ with real coefficients a, b, c, and a ≠ 0. The function y = emx is a solution if, and only if, m satisfies the auxiliary equation am2 + bm + c = 0. When the roots of this are the complex conjugates m = p ± iq, then y = e(p ± iq)x are complex solutions of (H). Nevertheless, real solutions are given by y = c1epx cos qx + c2epx sin qx.

Teknik Penentuan Solusi Sistem Persamaan Diferensial Linear Non-Homogen Orde Satu

Abstrak. Penentuan solusi sistem persamaan diferensial linear non-homogen orde satu dengan koefisien konstanta, dilakukan dengan mengubah sistem persamaan tersebut menjadi persamaan diferensial linear non homogen tunggal. Dari persamaan diferensial linear non homogen tunggal tersebut kemudian dicari solusi homogennya menggunakan akar-akar karakteristiknya, dan mencari solusi partikularnya dengan metode variasi parameter. Solusi umum dari persamaan diferensial linear tersebut adalah jumlah dari solusi homogen dan solusi partikularnya. Persamaan diferensial linear tunggal tersebut berorde- , yang solusi umumnya berbentuk . Selanjutnya dicari solusi umum berebentuk yang berkaitan dengan , solusi umum berbentuk yang berkaitan dengan dan , solusi umum berbentuk yang berkaitan dengan , , dan , demikian seterusnya sampai mencari solusi umum berbentuk yang berkaitan dengan , , , , . Kumpulan solusi umum yang berbentuk merupakan solusi umum dari sistem persamaan diferensial linear non homogen orde satu tersebut.Kata kunci: Diferensial, Linear, Non-Homogen, Orde, Satu. Technical to Find The System of Linear Non-Homogen Differential Equation of First OrderAbstract. Determination of first-order non-homogeneous linear differential equation system solutions with constant coefficients, carried out by changing the system of equations into a single non-homogeneous linear differential equation. From a single non-homogeneous differential equation, a homogeneous solution is then used using its characteristic roots, and looking for a particular solution with the parameter variation method. The general solution of these linear differential equations is the number of homogeneous solutions and their particular solutions. The single linear differential equation is n-order, the solution being in the form of . Then look for a general solution in the form of related to , a general solution in the form of related to and , general solutions in the form of related to , and , and so on until looking for a general solution in the form of related to , , , ..., . A collection of general solutions in the form of , , , ..., is the general solution of the first-order non-homogeneous linear differential equation system.Keywords: Linear, Differential, First, Order, Non-Homogeneous

Dynamic Analysis of Gravity Platform's Marine Riser under Wave Load

Gravity platform was widely used in marginal oilfield, so the strength and the stability of its marine riser under wave load became very important. By simplifying the Morison equation, the paper made the dynamic analysis of the riser come down to solving the fourth order non-homogeneous linear differential equation, thus got the analytical solution of its deflection, shear force and bending moment. And the comparison with Ansys simulation showed that this simplified method was feasible.

On the set of zero coefficients of a function satisfying a linear differential equation

Let K be a field of characteristic zero and suppose that f: → K satisfies a recurrence of the form \[ f(n) = \sum_{i=1}^d P_i(n) f(n-i), \] for n sufficiently large, where P1(z),. . .,Pd(z) are polynomials in K[z]. Given that Pd(z) is a nonzero constant polynomial, we show that the set of n ∈ for which f(n) = 0 is a union of finitely many arithmetic progressions and a finite set. This generalizes the Skolem–Mahler–Lech theorem, which assumes that f(n) satisfies a linear recurrence. We discuss examples and connections to the set of zero coefficients of a power series satisfying a homogeneous linear differential equation with rational function coefficients.