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A new perturbation method, multiple scales Lindstedt–Poincare (MSLP) is applied to jerk equations with cubic nonlinearities. Three different jerk equations are investigated. Approximate analytical solutions and periods are obtained using MSLP method. Both approximate analytical solutions and periods are contrasted with numerical and exact results. For the case of strong nonlinearities, obtained results are in good agreement with numerical and exact ones.

We consider the following two-dimensional system of non-linear equations: u ′ = g ⁢ ( t ) ⁢ | v | 1 α ⁢ sgn ⁡ v , v ′ = - p ⁢ ( t ) ⁢ | u | α ⁢ sgn ⁡ u , u^{\prime}=g(t)|v|^{\frac{1}{\alpha}}\operatorname{sgn}v,\quad v^{\prime}=-p(t% )|u|^{\alpha}\operatorname{sgn}u, where α > 0 {\alpha>0} , and g : [ 0 , + ∞ [ → [ 0 , + ∞ [ {g\colon{[0,+\infty[}\rightarrow{[0,+\infty[}} and p : [ 0 , + ∞ [ → ℝ {p\colon{[0,+\infty[}\rightarrow\mathbb{R}} are locally integrable functions. Moreover, we assume that the coefficient g is non-integrable on [ 0 , + ∞ ] {[0,+\infty]} . We establish new non-oscillation criteria for the considered system, which generalize known results for the corresponding linear system and for second order differential equations. In particular, the presented criteria are in compliance with the results of Hille and Nehari.

The linearization problem of a third-order ordinary differential equation by the tangent transformation is considered in the present paper. This is the first application of tangent (essentially) transformations to the linearization problem of third-order ordinary differential equations. Necessary and sufficient conditions for a third-order ordinary differential equation to be linearizable are obtained.

By the use of geometric methods for linearizing systems of second-order cubically semi-linear ordinary differential equations and the conditional linearizability of third-order quintically semi-linear ordinary differential equations, we extend to the fourth-order by differentiating the third-order conditionally linearizable equation. This yields criteria for conditional linearizability of a class of fourth-order semi-linear ordinary differential equations, which have not been discussed in the literature previously.

In this work we establish the theory of dichotomies for generalized ordinary differential equations, introducing the concepts of dichotomies for these equations, investigating their properties and proposing new results. We establish conditions for the existence of exponential dichotomies and bounded solutions. Using the correspondences between generalized ordinary differential equations and other equations, we translate our results to measure differential equations and impulsive differential equations. The fact that we work in the framework of generalized ordinary differential equations allows us to manage functions with many discontinuities and of unbounded variation.

In this paper, we propose an efficient numerical method for ordinary differential equation with fractional order, based on Legendre-Gauss-Radau interpolation, which is easy to be implemented and possesses the spectral accuracy. We apply the proposed method to multi-order fractional ordinary differential equation. Numerical results demonstrate the effectiveness of the approach.

A Pelczar, in [1], introduced and proved the existence and uniqueness of the iterative ordinary differential equations. M. Podisuk, in [2], introduced and proved the existence and uniqueness of the simple iterative ordinary differential equations. In this paper, the application of the simple iterative ordinary differential equations in the area of population growth will be illustrated and their results are shown in the examples.

One-dimensional oscillations are described by linear ordinary differential equations with closed form solutions extendable to forced oscillations. Besides the Lagrangian the Hamiltonian form of mechanics is introduced. Further applications are the motion of space frames (from Chapter 3) and two species models from population dynamics. By means of qualitative analysis (using phase portraits) the long time behavior can be analyzed. The principle of linearized stability reduces the stability question to linear systems, where it can be completely studied. Finally also variational problems and optimal control problems are discussed.

This chapter is an introduction to ordinary differential equations. The letters X, Y, Z will always denote Banach spaces, i.e., complete normed spaces.

We generalize a mathematical model of a bimolecular heterogeneous chemical reaction for the case of more than two reactants. The model is described by a finite system of parabolic and ordinary differential equations for which the existence and uniqueness theorem of a classical solution is proved.

In this paper, we establish a result of Leray-Schauder degree on the order interval which is induced by a pair of strict lower and upper solutions for a system of second-order ordinary differential equations. As applications, we prove the global existence of positive solutions for a multi-parameter system of second-order ordinary differential equations with respect to parameters. The discussion is based on the result of Leray-Schauder degree on the order interval and the fixed point index theory in cones.

Two autonomous, nonlinear, third-order ordinary differential equations whose dynamics can be represented by second-order nonlinear ordinary differential equations for the first-order derivative of the solution are studied analytically and numerically. The analytical study includes both the obtention of closed-form solutions and the use of an artificial parameter method that provides approximations to both the solution and the frequency of oscillations. It is shown that both the analytical solution and the accuracy of the artificial parameter method depend greatly on the sign of the nonlinearities and the initial value of the first-order derivative.

We consider fourth order quasilinear ordinary differential equations. Firstly, we classify positive solutions into four types according to their asymptotic properties. Then we derive existence theorems of positive solutions belonging to each type. Using these results, we can obtain an oscillation criterion, which is our main objective. Moreover, applying such criteria for ordinary differential equations to binary elliptic systems, we establish nonexistence theorems for positive solutions.

In this paper, we establish the product formula for the fixed point index on product cone, and then, as applications, consider the existence, nonexistence and multiplicity of positive solutions for a second-order ordinary differential system with parameters. The discussion is based on the product formula and the fundamental properties of the fixed point index.

We survey some results on ordinary differential equations in Banach spaces and pose several related open problems.

This research gives a complete Lie group classification of second-order delay ordinary differential equations of the form y ″ = f x , y , y τ , y ′ , y τ ′ , where τ > 0 is the delay, y τ = y ( x - τ ) and y τ ′ = y ′ ( x - τ ) . All classes of second-order delay ordinary differential equations admitting a Lie algebra are presented in the manuscript. The method for finding the solution for this type of differential equation is developed.

A model of coupled parabolic and ordinary differential equations for a heterogeneous catalytic reaction is considered and the existence and uniqueness theorem of the classic solution is proved.

We establish asymptotic representations for one class of solutions of two-dimensional systems of ordinary differential equations that are more general than systems of the Emden–Fowler type.

We provide linearizability criteria for a class of systems of two third-order ordinary differential equations that is cubically nonlinear in the first derivative, by differentiating a system of second-order quadratically nonlinear ordinary differential equations and using the original system to replace the second derivatives. The procedure developed splits into two cases: those for which the coefficients are constant and those for which they are variables. Both cases are discussed and examples given.

For higher order ordinary differential equations, new sufficient conditions on the existence and uniqueness of periodic solutions are established. Results obtained cover the case when the right-hand side of the equation is not of a constant sign with respect to an independent variable.

Solution of linearization problem of fourth-order ordinary differential equations via contact transformations is presented in the paper. We show that all fourth-order ordinary differential equations that are linearizable by contact transformations are contained in the class of equations which is at most quadratic in the third-order derivative. We provide the linearization test and describe the procedure for obtaining the linearizing transformations as well as the linearized equation. Moreover, we obtain the general form of ordinary differential equations of order greater than four linearizable via contact transformations.

In this paper, the basic theory for the initial value problems for fractional functional differential equations is considered, extending the corresponding theory of ordinary functional differential equations.

Homotopy perturbation method is used to solve specific second order ordinary differential equations and tested for different examples. The results obtained demonstrate efficiency of the proposed method.

Conditions ensuring that solutions of ordinary differential equations have vertical asymptotes are obtained. Bibliography: 8 titles.

In this paper, we introduce the notion of generalized weakly ?-contractive mappings and give some common fixed point results for this type of contraction. The presented theorems extend, generalize and improve many existing results in the literature. Two examples as well as an application to ordinary differential equations are also given in order to illustrate the effectiveness of the obtained results. © 2016 Miskolc University Press.

This chapter introduces the basic techniques of scaling and the ways to reason about scales.

We will give concrete representation formulas of the approximate solutions for linear ordinary differential equations following the Tikhonov regularization and using a finite number of data.

For the need of the theory of oscillatory solutions of dissipative abstract equations a suitable function is constructed as a solution of the second order ordinary differential equation with nonlinearity in the first derivative.

We consider an implicit first-order ordinary differential equation with complete integral. In [3], the authors give a generic classifications of first-order ordinary differential equations with complete integral with respect to the equivalence relation which is given by the group of point transformations. The classification problem is reduced to the classification of a certain class of divergent diagrams of mapping germs. In this paper, we give a generic classifications of bifurcations of such differential equations as an application of the Legendrian singularity theory.

We obtain a criterion for the existence of solutions of degenerate inhomogeneous Fredholm boundary-value problems for a system of ordinary differential equations under the assumption that the degenerate system of differential equations can be reduced to the central canonical form. The results are illustrated by examples.

A new algorithm for linearization of a third-order ordinary differential equation is presented. The algorithm consists of composition of two operations: reducing order of an ordinary differential equation and using the Lie linearization test for the obtained second-order ordinary differential equation. The application of the algorithm to several ordinary differential equations is given.