This paper investigates the transient solution of discrete and continuous time Markov chains (DTMC and CTMC) using the class of methods based on the undetermined coefficients approach. Two methods that belong to this class are investigated: The well known Eigenvectors method, and what it is referred in the paper as the Vandermonde method, because it leads to the solution of a Vandermonde system of equations. Even if the Vandermonde method is possibly the simplest method to obtain the transient solution of a Markov chain, it has received few attention in the literature. This paper fills this gap, showing its advantages. The Vandermonde method is also exploited to derive interesting relations between a CTMC and its uniformized chain. Based on them, it is proposed a simple, yet powerful method to compute the transient solution of Markov chains referred to as the Uniformized Vandermonde method.