Abstract: This paper introduces a procedure to apply Laplace transform theory in solving a class of Sturm – Liouville boundary – value problems, namely those having polynomial coefficients. The method demonstrates that not only can Laplace transforms solve differential equations satisfying only initial-value conditions, but can also be adaptable to solving boundary – value problems. In the end, the application of inverse Laplace transforms to the problem introduces the reader to a novel approach in the analysis and solution of a specific class of Sturm – Liouville boundary problems.