Planar linear differential weakly delayed systems with constant matrices and equivalent ordinary differential systems

Considered is a linear planar delayed differential system $\dot{x}\left(t\right)=Ax\left(t\right)+Bx(t-\tau )$, where $t\ge 0$, $\tau >0$ is a constant delay and A, B are $2\times 2$ constant matrices. Assuming that the system is weakly delayed, its general solution is constructed utilizing the Laplace transform. All the cases are specified of the solutions merging. Moreover, ordinary differential systems are considered such that general solutions of both delayed and non-delayed systems coincide when a transient interval is passed. Initial data for the relevant non-delayed systems are used such that these define the same solution as the corresponding initial data to the delayed system. An analysis of previous findings is given with two illustrative examples considered. Some open problems are suggested as well.