On a General Class of Second-Order, Linear, Ordinary Differential Equations Solvable as a System of First-Order Equations

Abstract

Abstract. An approach for solving general second-order, linear, variable-coefficient ordinary differential equations in standard form under initial-value conditions is presented for the case of a specific constant-form relation between the two otherwise arbitrary coefficients. The resulting system of linear equations produces fundamental (or state transition) matrix elements used to create integral-and closed-form solutions for both homogeneous and nonhomogeneous differential equation variants. Two example equations are chosen to illustrate application. A short discussion is presented on the comparison of the theoretical results for these examples with the corresponding symbolic integration outputs provided by several commercial programs which were seen, at times, to be long and unwieldy or even non-existent.Mathematics subject classification (2010): 34A30, 93C15.

Feedback Diagram Application for the Generation and Solution of Linear Differential Equations Solvable by Quadrature

“…x , respectively. Note that the former case of constant a 11 = a has been previously analyzed from a related yet different viewpoint in [1]. Similarly, other choices, such as a 11 (x) = tanh(x) or − tan(x), also lead to comparable coefficient interrelationships and equation solutions.…”

“…The feedback diagram of Figure 1 presents the usual depiction of the second-order, linear, differential Equation (1). It includes two integrators, a summing box on the left, two amplifiers representing the variable coefficients ( ) and ( ) , and the nonhomogeneous driving function ( ).…”

“…u(x) = A I (x)u(x) + b f y (x). (6) Matrix A I (x) is the companion matrix of the corresponding characteristic polynomial of Equation (1), and the standard general solution to equations (5) and (6) [11] (pp. 114-118), [12] (pp.…”

“…In Control and Systems Theory, feedback diagrams have long been used to provide clear pictorial representations of closed-loop systems and their corresponding differential equation descriptions. These diagrams have proven to be useful tools in the analysis of differential equations for both characterization and solutions [1]. In this paper, we construct and apply a method derived from competing diagrammatic representations to investigate the initial value problem of the linear second-order, variable coefficient, nonhomogeneous, differential equation,…”

“…We are simultaneously seeking quadrature, or integral form, solutions, as well as the functional forms of the coefficients p(x) and q(x), for which these solutions are valid. To achieve this purpose, two second-order linear systems are presented: The first portraying the system under investigation described by Equation (1) and the second modified to provide integral form general solutions. The coefficients of this second system are then adjusted to correspond to those of the first, thereby making the two systems effectively identical and providing solutions for both.…”