Hilbert Number for a Family of Piecewise Nonautonomous Equations

For family \(x'=(a_0+a_1\cos t+a_2 \sin t)|x|+b_0+b_1 \cos t+b_2 \sin t\) $x^{\prime }=(a_0+a_{1}cost+a_{2}sint)|x|+b_0+b_{1}cost+b_{2}sint$ , we solve three basic problems related with its dynamics. First, we characterize when it has a center (Poincaré center focus problem). Second, we show that each equation has a finite number of limit cycles (finiteness problem), and finally we give a uniform upper bound for the number of limit cycles (Hilbert problem 16).