THE purpose of this note is to derive an analytic solution to the second order linear differential equation whose coefficients vary exponentially with time. Such equations arise in the investigation of small perturbations in the linear motion of a body through the atmosphere in which the density may be considered to vary exponentially with height. For example, if the perturbation considered be a small angle of incidence of the body to the direction of the undisturbed motion and the velocity of descent is taken to be constant, then with linearized aero‐dynamics both the restoring and damping moments are proportional to the atmospheric density and therefore vary exponentially with height and hence with time. For sufficiently slow time rates of variation in the coefficients an iterative solution may be obtained in the form of an infinite series. The difficulty inherent in this method lies in determining the convergence of the scries obtained. Apart from this, the method fails in the topically interesting case of rapid descent where the coefficients vary rapidly with time. The following method overcomes this difficulty and leads to a solution valid over the whole range of descent velocity, provided only that the motion due to perturbation is lightly damped, a condition usually satisfied in practical cases.
Cuming, H. (1957), "Perturbations of a Body in an Exponential Atmosphere: A Solution of the Equation arising in the case of Small Angular Displacements", Aircraft Engineering and Aerospace Technology, Vol. 29 No. 4, pp. 123-124. https://doi.org/10.1108/eb032816Download as .RIS
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