Abstract: A new exponential matrix function is introduced to obtainthe special solutions, based on Duhamel convolution integration for thenon-homogenous polynomial terms of the first order constant coefficientordinary equation, exponential functions or their products., Based on thenew exponential matrix, an efficient recursive algorithm is proposed in thispaper with a precise integration method (PTM) applied to the specialsolution. The processes of PTM for general solution and the proposed methodfor special solution are closely integrated. This combination is confirmedto have a high computational efficiency.. The proposed method is alsovalidated to have more convenient and more applied range. A general formulais given to greatly save computation time. Numerical examples are alsodemonstrated the validity and efficiency of the method.