Explicit solutions and solution merging in WD planar differential systems

The paper studies weakly delayed planar linear differential systems [Formula: see text] with constant matrices [Formula: see text], [Formula: see text], [Formula: see text], and with constant delays [Formula: see text] and [Formula: see text], [Formula: see text]. The definition of weakly delayed systems is analyzed and coefficient criteria for entries of matrices [Formula: see text], [Formula: see text] and [Formula: see text] are derived ensuring that the system is weakly delayed in two different cases when [Formula: see text] and [Formula: see text]. Explicit formulas are constructed for solving the initial problem and for finding the general solutions, depending on the eigenvalues of the matrix [Formula: see text]. It is shown that, after a transient interval is passed, the behavior of the solutions can be described by a combination of two exponential-type functions. Conditions are also derived for solution merging. Most of the results are proved by applying the Laplace transform technique. Previously known results are mentioned and compared.