On limit cycles of piecewise differential systems formed by arbitrary linear systems and a class of quadratic systems

. We study the continuous and discontinuous planar piecewise diﬀerential systems separated by a straight line and formed by an arbitrary linear system and a class of quadratic center. We show that when these piecewise diﬀerential systems are continuous, they can have at most one limit cycle. However, when the piecewise diﬀerential systems are discontinuous, we show that they can have at most two limit cycles, and that there exist such systems with two limit cycles. Therefore, in particular, we have solved the extension of the 16th Hilbert problem to this class of diﬀerential systems.