Abstract In this paper, we obtain all the locally convex plane curves with proportional curvatures (Euclidean and affine). The analysis is performed from a special second order differential equation that, in appropriated coordinates, and using the support function the problem is reduced to the properties of a periodic linear nonhomogeneous second differential equation of the type y''+y=a (t), \; a (t+2) =a (t) y ′ ′ + y = a (t), a (t + 2 π) = a (t).
Garcia et al. (Mon,) studied this question.