We apply the averaging method of first order to study the maximum number of limit cycles of the ordinary differential systems of the form   ¨x + x = ε (f1(x, y)y + f2 (x, y)) , ¨y + y = ε (g1(x, y)x + g2 (x, y)) , where f1(x, y) and g1(x, y) are real cubic polynomials; f2(x, y) and g2(x, y) are real quadratic polynomials. Furthermore ε is a small parameter.