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.
A novel method for generating and providing quadrature solutions to families of linear, second-order, ordinary differential equations is presented in this paper. It is based upon a comparison of control system feedback diagrams—one representing the system and equation under study and a second equalized to it and providing solutions. The resulting Riccati equation connection between them is utilized to generate and solve groups of equations parameterized by arbitrary functions and constants. This method also leads to a formal solution mechanism for all second-order linear differential equations involving an infinite series of integrals of each equation’s Schwarzian derivative. The practicality of this mechanism is strongly dependent on the series rates of and allowed regions for convergence. The feedback diagram method developed is shown to be equivalent to a comparable method based on the differential equation’s normal form and another relying upon the grouping of terms for a reduction of the equation order, but augmenting their results. Applications are also made to the Helmholtz equation.
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