Integrating factors for second-order ODEs
In connection with second-order ordinary differential equations (ODEs), the authors present a systematic algorithm for determining the existence of integrating factors and their explicit form, when they have the forms \(\mu (x,y), \mu(x,y')\) and \(\mu(y,y')\). The algorithm can determine the existence and explicit form of the integrating factors themselves without solving any differential equations, except for a linear ODE in one subcase of the \(\mu(x,y)\) problem. Examples of ODEs not having point symmetries are shown to be solvable using the algorithm. The scheme was implemented in Maple, in the framework of the \textsl{ODEtools} package and its ODE-solver. A comparison between this implementation and other computer algebra ODE-solvers in tackling nonlinear examples from \textit{E. Kamke's} book is shown [Differentialgleichungen. Lösungsmethoden und Lösungen. 1. Gewöhnliche Differentialgleichungen (1942; Zbl 0026.31801)].
second-order ordinary differential equations
scientific article