1 On sinusoidal periodic solution of mixed Lienard type equations K. K. D. Adjaï 1 , E. A Doutètien 1 , J. Akande 1 , M.D. Monsia 1* 1- Department of Physics, University of Abomey-Calavi, Abomey-Calavi, 01.BP.526, Cotonou, BENIN. Abstract We study in this paper the existence of exact periodic solutions of the mixed Lienard type equations. We show for the first time the conditions to ensure the exact and explicit integrability and to obtain sinusoidal periodic solution. As a result, the equation can be used to describe harmonic and isochronous oscillations of dynamical systems. Keywords: Mixed Lienard type equations, sinusoidal periodic solution, harmonic and isochronous oscillations. Introduction A well studied equation in the literature is the mixed Lienard type differential equation [1-3] 0 ) ( ) ( ) ( 2     x h x x x x u x      (1) where ) ( x u , ) ( x  and ) ( x h are arbitrary function of x , and the overdot designates the differentiation with respect to time. The equation (1) contains several classes of differential equations. When 0 ) (  x u , the equation (1) becomes 0 ) ( ) (    x h x x x     (2) which has been for a long time investigated in the literature [4-9]. An important result is that the equation (2) can exhibit not only periodic solution but also isochronous property under some conditions [5, 7-9]. It is also observed that for some functions ) ( x  and ), ( x h the equation (2) has sinusoidal solution as the linear harmonic oscillator [8]. A general sinusoidal solution has been ensured for the equation (2) in [9] for some choice of ) ( x  and ) ( x h . It is for the first time these exceptional results are obtained for a dissipative equation of type (2). Putting , 0 ) (  x u and 0 ) (  x  , yields the equation * Corresponding author : E-mail : monsiadephin@yahoo.fr