A class of second-order differential equations and related first-order systems

A class of second-order nonlinear differential equations which arises in several branches of mathematical physics is considered. It is shown that equations of this class may be factorised into first-order equations of 'Riccati type'. Conditions are obtained, on the coefficient functions of the second-order equations, for the first-order equations to be of matrix Riccati form, whose solutions have a finite superposition property. The factorisation into first-order equations is then not unique, and there is an alternative first-order set of equations whose solutions do not have this superposition property. A second-order equation arising in the theory of pellet fusion processes is investigated in detail. Solutions are obtained when the corresponding first-order equations are of matrix Riccati from and shown to be equivalent to solutions derived by alternative methods. Lagrangian systems giving rise to equations of the class are also considered.