Using the tools of optimal control, semiconvex duality and max-plus algebra, this work derives a unifying representation of the solution for the matrix differential Riccati equation (DRE) with time-varying coefficients. It is based upon a special case of the max-plus fundamental solution, first proposed in Fleming and McEneaney (2000). Such a fundamental solution can extend a particular solution of certain bivariate DREs into the general solution, and the DREs can be analytically solved from any initial condition. This paper also shows that under a fixed duality kernel, the semiconvex dual of a DRE solution satisfies another dual DRE, whose coefficients satisfy the matrix compatibility conditions involving Hamiltonian and certain symplectic matrices. For the time-invariant DRE, this allows us to make dual DRE linear and thereby solve the primal ORE analytically. This paper also derives various kernel/duality relationships between the primal and time shifted dual DREs, which lead to an array of DRE solution methods. Time-invariant analogue of one of these methods was first proposed in McEneaney (2008). (C) 2011 Elsevier Ltd. All rights reserved.