We consider limit cycles of a class of polynomial differential systems of the form ẋ = y, where β and α are positive integers, g 2j and f 2j have degree m and n, respectively, for each j = 1, 2, and ε is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center ẋ = y, ẏ = −x using the averaging theory of first and second order.