This paper deals with optimization of a class of linear time-varying dynamic systems with n states and m control inputs commanded to move between two fixed states in a prescribed final time. Using a conventional procedure with Lagrange multipliers, it is well known that the optimal solution must satisfy 2n first-order linear differential equations in the state and Lagrange multiplier variables. Due to the specific nature of boundary conditions with states given at the two end times, the two-point boundary value problem must be solved iteratively using shooting methods, that is, there is no closed-form quick procedure to obtain the solution of the problem. In this paper, a new procedure for dynamic optimization of this problem is presented that does not use Lagrange multipliers. In this new procedure, it is shown that for a dynamic system with n = pm, where p is an integer, the optimal solution must satisfy m 2p-order differential equations. Due to the absence of Lagrange multipliers, the higher order differential equations can be solved efficiently using classical weighted residual methods.