general solution of first order homogeneous differential equation

... "07 General Solution of the One-Dimensional Wave Equation" (2004) Foundations of Wave Phenomena Book 16. http://digitalcommons.usu.edu/foundation_wave/16 Trang 27 General Solution of the ... — which provides every possible solution to the wave equation.* We can find the general solution of the (one-dimensional) wave equation as follows Make a change of variables t = t(s, u), x = x(s, ... altering the solution to the wave equation; this redundancy is equivalent to the arbitrariness of x0 (exercise) We have now obtained the general solution to the wave equation expressed in terms of any

Ngày tải lên: 25/10/2022, 08:48

... existence of at least one and at least three positive solutions to problem 1.5 For the case of α 2, boundary value problem 1.5 is related to a boundary value problems of integer-order differential equation ... Difference Equations 13 Conclusions In this paper, by using the fixed point theorem of cone, we have investigated the existence of positive solutions for a class of nonlinear fractional differential equations ... and E Mboumi, “Positive solutions of a boundary value problem for a nonlinear fractional differential equation,” Electronic Journal of Qualitative Theory of Differential Equations, no 3, pp 1–11,

Ngày tải lên: 21/06/2014, 05:20

... and Differential EquationsNguyen Thieu Huy Hanoi University of Science and Technology Trang 23.3 Solution: To solve Equation (1), first solve the equationsTrang 33.4 Integrating Factors: In general, ... the (differential) equations. Analysis of the Model (A) To solve equations (B) To find the properties of solutions (C) If it is too difficult to solve the equations, try to approximate the solutions ... The solution is yx = C 4 Linear Equations 4.1 Definition A first-order linear differential equation has the form y0+ p(x )y = q(x ) Example Looking back: dvdt + kmv = g ; dtdI +RLI = EL This equation

Ngày tải lên: 23/04/2020, 08:00

... theorems on oscillation and nonoscillation of solutions to 1.3 are obtained Thus, solutions of two different equations and/or twodifferent solutions of the same equation are compared, which allows to ... following first-order dynamic equation:t0,∞Tprovided that−A ∈ Rt0,∞T,R see 4, Theorem 2.71 Thus, the general solution of the initial value problem for the nonhomogeneous equation x ... Article Nonoscillation of First-Order Dynamic Equations with Several Delays Elena Braverman1 and Bas¸ak Karpuz2 1 Department of Mathematics and Statistics, University of Calgary, 2500 University

Ngày tải lên: 21/06/2014, 11:20

... introduction of the basic theory of impulsivedifferential equations, see Lakshmikantham et al 1; for an overview of existing results and of recent research areas of impulsive differential equations, ... positive solution y t, t ∈ J with property 3.17. Proof The proof is similar to that of Theorem 3.2 of6 Theorem 3.5 Assume that H1 , H2, 3.1 of H3 and 3.14 and 3.15 of H4 hold ... paper, we consider the existence ofpositive solutions for a class of boundary value problems with integral boundary conditions of fourth-order impulsive differential equations: k represent the right-hand

Ngày tải lên: 21/06/2014, 16:20

... ω-periodic solution of1.3 or 1.4 if it satisfies this equation almost everywhere onR For the case τ k t ≡ tk  1, , n, the problem on the existence and uniqueness of an ω-periodic solution of1.4 ... If the function u is an ω-periodic solution of1.3, then its restriction to 0, ω is a solution of problem 2.26, 2.27, and vice versa, if u is a solution of problem2.26, 2.27, then its ... 2.31 Let u be a solution of problem2.30, 2.27 for some λ ∈0, 1 Then its periodic extension toR with a period ω is a solution of2.32, and according to one of the conditions of the lemma,

Ngày tải lên: 21/06/2014, 22:20

... oscillation of all solutions of first-order delay dynamic equations on arbitrary time scales, hence combining and extending results for corresponding differential and difference equations Examples, some of ... oscillation criterion for first order dynamic equations,” to appear in Functional Differential Equations. 10 Y S¸ahiner and I P Stavroulakis, “Oscillations of first order delay dynamic equations,” Dynamic ... criteria for first order delay dynamic equations,” Far East Journal of Applied Mathematics, vol 18, no 3, pp 289–304, 2005. 8 B G Zhang and X Deng, “Oscillation of delay differential equations

Ngày tải lên: 22/06/2014, 11:20

... lie between a lower and an upper solution of suitable operators Our purpose is to ensure the stability of at least one periodic solution of a first-order difference equation We will prove such result ... optimality of the obtained results 2 Preliminaries This paper is devoted to study the stability, by using the method of lower and upper solutions, of theN-periodic solutions of the following first-order ... more precise description of the set of solutions of problem (P N). Lemma 2.5 Assume conditions ( H) and (H f ) Let u, v ∈[α,β] be two solutions of problem (P N) Then, one of the following statements

Ngày tải lên: 23/06/2014, 00:20

... Ratio of the height of the base of a first-order branch to tree height BE cm year –1 Elongation of a first-order branch per year FBbmI g year –1 Increment in foliar biomass of a first-order branch ... in terms of tree height per year B year –1 Birth rate of first-order branches per tree per year D year –1 Death rate of first-order branches per tree per year ∆N year –1 Change in first-order branch ... observation of first-order branches We use a simple index to express the competitive status of individual trees, and report 1) the patterns in growth of individuals, 2) population dynam-ics of first-order

Ngày tải lên: 08/08/2014, 14:22

... rates of population and depopulation of an atomic-energy level of hydrogen We express this equation of statistical equilibrium (see Aller 1956; Menzel 1962) as where the right-hand side of equation ... usefulness of the differential-equation method is greatly decreased if the b,'s must be included in the cascade term and if An in excess of unity is used, the general iterative method of solution ... assumptions, he reduces equation (7) to a second-order differential equation which can be solved to determine the bn's I n 8 IV we will discuss a more general iterative process for solving equation (7)

Ngày tải lên: 23/10/2022, 15:17

... of first-order partial differential equations, including questions such as estimates of solutions of initial problems, estimates of domains of solutions, estimates of the difference between solutions ... 2functions of one variable, which are solutions of ordinary differential equations orinequalities There exist many generalizations of the above classical results We list some of them below Differential ... Functional differential inequalities, Haar pyramid, Comparison theorems, Weak solutions of initial problems 1 Introduction Two types of results on first-order partial differential or functional differential

Ngày tải lên: 21/06/2014, 02:20

... pagesdoi:10.1155/2011/483057 Research Article Iterative Solutions of Singular Boundary Value Problems of Third-Order Differential Equation Peiguo Zhang Department of Elementary Education, Heze University, ... boundary value problem of third-order differential equations,” Mathematica Slovaca, vol 60, no 4, pp 485–494, 2010 4 Y Sun, “Positive solutions for third-order three-point nonhomogeneous boundary ... “Existence of solutions for a class of third-order nonlinear boundary valueproblems,” Journal of Mathematical Analysis and Applications, vol 294, no 1, pp 104–112, 2004. 14 D Guo, Semi-Ordered

Ngày tải lên: 21/06/2014, 05:20

... notions of the fractional calculus and the theory of the operators for integration and differentiation of fractional order.Section 3is devoted to the study of the existence and uniqueness of positive ... positive solution ut ∈ S. Trang 94 Maximal and Minimal Solutions TheoremIn this section, we consider the existence of maximal and minimal solutions for1.2 Definition 4.1 Let mt be a solution of ... 2 E Buckwar and Y Luchko, “Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations,” Journal of Mathematical Analysis and Applications,

Ngày tải lên: 21/06/2014, 07:20

... introduction of the basic theory of impulsive differential equations, see Lakshmikantham et al 1; for an overview of existing results and of recent research areas of impulsive differential equations, ... obtained the existence of positive solutions by means of fixed point index theory Recently, Yang and Wei 23 and the author of 24 improved and generalized the results of Pang et al.22 by ... investigating the existence of positive solutions of nth impulsive differential equations by using the fixed point theorem of cone expansion and compression The objective of the present paper is to

Ngày tải lên: 21/06/2014, 16:20

... these phenomena The analysis starts with a second-order differential equation, free of constants, that offers a general way of describ-ing them This equation is then integrated and applied to illustrative ... phenomena, the initial slope of the data plot and the limiting value at saturation A first-order differential equation for the slope emerged that led to the concept of the effective binding rate ... the first-order D.E for the slope,Integrating again and rearranging gives This algebraic relation, when substituted into equation (1), satisfies the second-order D.E Therefore, it is a general solution

Ngày tải lên: 13/08/2014, 16:20

... ð2Þ Proof Suppose that y is solution of (1), then by the above definition, we state thatgðt,yðtÞÞyðtÞ is continuous Applying the ABC fractional derivative of orderα, we retrieve the first equation ... then the equation y = A1yA2yadmits a solution in S Let us start this section by defining what we mean by a solution of problem (1) Definition The function y CðJ,ℝÞ is said to be a solution of problem ... paper deals with a nonlinear hybrid differential equation written using a fractional derivative with a Mittag–Leffler kernel Firstly, we establish the existence of solutions to the studied problem

Ngày tải lên: 29/03/2021, 16:41

... differential equation theory Many results about the existence of periodic solutions for second-order differential equations have been obtained by combining the classical method of lower and upper solutions ... and uniqueness of a periodic solution for second-order ordinary differential equations with some assumption 1 Introduction The study on periodic solutions for ordinary differential equations is a ... pagesdoi:10.1155/2011/192156 Research Article Existence and Uniqueness of Periodic Solution for Nonlinear Second-Order Ordinary Differential Equations Jian Zu College of Mathematics, Jilin University, Changchun 130012,

Ngày tải lên: 14/07/2023, 09:44

... Every solution x of (4) satisfies α ≤ x ≤ β on I and, therefore, it is a solution of (1) in [α, β] First, notice that if x is a solution of (4) then p(·, x(·)) ∈ [α, β] Hence the definition of lower ... with a second-orderequation Therefore, we are obliged to distinguish between the concepts of minimal solution and least solution (or maximal and greatest solutions),unfortunately often identified ... maximal solutions to first-order differential equations with state-dependent deviated argumentsRub´en Figueroa∗ and Rodrigo L´opez Pouso Department of Mathematical Analysis, University of Santiago

Ngày tải lên: 20/06/2014, 20:20

... inequalities for a class ofeven-order differential equations Qi-Ming Zhang∗1 and Xiaofei He2 1College of Science, Hunan University of Technology, Zhuzhou, Hunan 412000, P.R China 2College of Mathematics ... a class of even-order differential equations Journal of Inequalities and Applications 2012, 2012:5 doi:10.1186/1029-242X-2012-5 Qi-Ming Zhang (zhqm20082008@sina.com) Xiaofei He (hexiaofei525@sina.com) ... criterion of Lyapunov Am J Math 73, 885–890 (1951) [3] Cheng, SS: Lyapunov inequalities for differential and difference equations Fasc Math 23, 25–41 (1991) [4] He, Z: Existence of two solutions of

Ngày tải lên: 20/06/2014, 21:20

... zero as t ® ∞ Proof The proof of the corollary is similar to the proof of Corollary 3.3 and so it is omitted From the above results on the oscillation of odd-order differential equation and Lemma ... Oscillation of even-order neutral delay differential equations Adv Differ Equ 2010, 1 –9 (2010) 14 Li, T, Thandapani, E: Oscillation of solutions to odd-order nonlinear neutral functional differential equations ... Oscillation of solutions of non-linear neutral delay differential equations of higher order for p(t) = ± 1 Arch Math 40, 359–366 (2004) 16 Tang, S, Li, T, Thandapani, E: Oscillation of higher-order

Ngày tải lên: 20/06/2014, 22:20