Higher-Order Linear Differential Equations

In this paper, we define sufficient conditions for the solvability in quadratures of systems of linear ordinary differential equations of higher orders with variable coefficients. The method of variation of constants for equations of higher orders and for systems of equations of first order is generalized to the case of such systems. Results are obtained by reducing the systems under consideration to matrix equations of higher orders studied by methods of the theory of Lie algebras.

CHAPTER II - LINEAR DIFFERENTIAL EQUATIONS. SUPPLEMENTARY REMARKS ON THE THEORY OF DIFFERENTIAL EQUATIONS

In this paper, we consider characterisations of the class of unitary matrix integrals $\big\langle (\det U)^q {\rm e}^{s^{1/2} \operatorname{Tr}(U + U^\dagger)} \big\rangle_{U(l)}$ in terms of a first-order matrix linear differential equation for a vector function of size $l+1$, and in terms of a scalar linear differential equation of degree ${l+1}$. It will be shown that the latter follows from the former. The matrix linear differential equation provides an efficient way to compute the power series expansion of the matrix integrals, which with $q=0$ and $q=l$ are of relevance to the enumeration of longest increasing subsequences for random permutations, and to the question of the moments of the first and second derivative of the Riemann zeta function on the critical line, respectively. This procedure is compared against that following from known characterisations involving the $\sigma$-Painlevé III$'$ second-order nonlinear differential equation. We show too that the natural $\beta$ generalisation of the unitary group integral permits characterisation by the same classes of linear differential equations.

In this work we consider rational ordinary\n\t\t\t\t differential equations dy/dx = Q(x, y)/P(x, y), with Q(x, y)\n\t\t\t\t and P(x, y) coprime polynomials with real coefficients. We\n\t\t\t\t give a method to construct equations of this type for which a\n\t\t\t\t first integral can be expressed from two independent solutions\n\t\t\t\t of a second-order homogeneous linear differential equation.\n\t\t\t\t This first integral is, in general, given by a non Liouvillian\n\t\t\t\t function.\n\t\t\t\t We show that all the known families of quadratic systems\n\t\t\t\t with an irreducible invariant algebraic curve of arbitrarily high\n\t\t\t\t degree and without a rational first integral, can be constructed\n\t\t\t\t by using this method. We also present a new example of this\n\t\t\t\t kind of family.\n\t\t\t\t We give an analogous method for constructing rational\n\t\t\t\t equations but by means of a linear differential equation of\n\t\t\t\t first order.

For nonlinear difference equations of higher order which satisfy certain positivity conditions, in particular an inequalilty involving the partial derivatives of the defining function, three kinds of global stability are considered. In case the difference equation is autonomous, conditions for stability trichotomy are given. A difference equation is said to possess stability trichotomy if either all (non-zero) solutions are unbounded or all solutions converge to zero or all (non-zero) solutions converge to a positive equilibrium. In case of a non-autonomous difference equation it is shown that under certain conditions the solutions, even if highly erratic, exhibit path stability. Path stability means that a solution when being disturbed comes finally back to its original behavior. For an autonomous and homogeneous difference equation with possibly unbounded solutions, conditions are specified under which the solutions are relative stable, which means that for any (non-zero) solution the ratio of any two s...

Higher-order linear differential equations have great significance theoretically and practically. This paper solves a third-order linear differential equation with constant coefficients by method of undetermined coefficients. Its strengths and limitations were analysed. Analysis suggests that the method of undetermined coefficients requires that the inhomogeneous term be algebraic, exponential, or sinusoidal functions only. Additionally, if the inhomogeneous term is a solution of the differential equation itself, may yield a linear combination of . In such case, the prediction for will be modified by multiplying the variable however many times necessary.

The paper talks about solution of linear homogeneous fuzzy differential equation of first order, under fuzzy initial condition.Here symmetrical triangular and trapezoidal fuzzy numbers are used as initial conditions of the said differential equation and using concept of -cut interval arithmetic a general solution to the problem is proposed.

The main aim of this paper is to investigate various types of Ulam stability and Mittag-Leffler stability of linear differential equations of first order with constant coefficients using the Aboodh transform method. We also obtain the Hyers–Ulam stability constants of these differential equations using the Aboodh transform and some examples to illustrate our main results are given.

The purpose of this paper is to study the Hyers–Ulam stability and Generalized Hyers–Ulam stability of the general Linear Differential Equations of first order and second order with constant coefficients using Fourier Transform method. Moreover, the Hyers–Ulam stability constants of these differential equations are obtained. Some examples are given to illustrate the main results.

The purpose of this paper is to investigate the use of rational Chebyshev (RC) collocation method for solving high-order linear ordinary differential equations with variable coefficients. Using the rational Chebyshev collocation points, this method transforms the high-order linear ordinary differential equations and the given conditions to matrix equations with unknown rational Chebyshev coefficients. These matrices together with the collocation method are utilized to reduce the solution of higher-order ordinary differential equations to the solution of a system of algebraic equations. The solution is obtained in terms of RC functions. Numerical examples are given to demonstrate the validity and applicability of the method. The obtained numerical results are compared with others existing methods and the exact solution where it shown to be very attractive and maintains better accuracy.

In this paper the order and the hyper-order of the solutions of higher-order homogeneous linear differential equations is investigated.

On inequalities of Lyapunov type

The present paper is dedicated to the investigation of optimal control problems whose behavior is described by quasilinear differential equations of first order on the plane with nonlocal Bitsadze–Samarski boundary conditions. A theorem of the existence and uniqueness of a generalized solution in the space \(C_{\mu }(\overline{G})\) is proved for quasilinear differential equations; necessary conditions of optimality are obtained in terms of the principle of maximum; the Bitsadze–Samarski boundary value problem is proved for a linear differential equation of first order; the existence of a solution in the space \(C_{\mu }^{p}(\overline{G})\) is proved and an a priori estimate is derived. A theorem on the necessary and sufficient condition of optimality is proved for a linear optimal control problem.

Variation of parameters and the method of Kryloff and Bogoliuboff

Within the framework of the kinetic approach, the authors have solved the problem on rarefied-gas flow in a cylindrical duct in the presence of the longitudinal temperature gradient. The Williams kinetic equation was used as the basic equation, and the diffuse-reflection model, as the boundary condition on the duct wall. This enabled the authors to consider the solution of the problem in linearized form. To find a linear correction to the locally equilibrium distribution function, the problem was reduced to solution of a linear homogeneous partial differential equation of first order. A solution to the latter was constructed with the method of characteristics. With account taken of the obtained solution and on the basis of the statistical meaning of the distribution function of gas molecules by coordinates and velocities, the authors constructed the profile of the heat-flux vector in the duct and computed the heat flux through the duct cross section. A numerical analysis of final expressions was made. A comparison with analogous results obtained with the discrete-ordinates method has shown that the solution procedure proposed in the work leads to correct results in a wide range of values of the duct radius.