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

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 integraland 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.