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This article deals with the asymptotic behavior of fourth order differentialequation where the coefficients are perturbations of linear constantcoefficient equation. We introduce a change of variable and deduce that the newvariable satisfies a third order differential equation of Riccati type. Weassume three hypothesis. The first is the following: all roots of thecharacteristic polynomial associated to the fourth order linear equation hasdistinct real part. The other two hypothesis are related with the behavior ofthe perturbation functions. Under this general hypothesis we obtain four mainresults. The first two results are related with the application of fixed pointtheorem to prove that the Riccati equation has a unique solution. The nextresult concerns with the asymptotic behavior of the solutions of the Riccatiequation. The fourth main theorem is introduced to establish the existence of afundamental system of solutions and to precise formulas for the asymptoticbehavior of the linear fourth order differential equation.

Asymptotic Integration of a Linear Fourth Order Differential Equation of Poincar\'e Type