Comparison of the Classical Method and the Eigenvalue–Eigenvector Approach in the Analytical Solution of Two-Compartment Pharmacokinetic Models

In this study, two fundamental analytical approaches used in solving linear homogeneous differential equation systems with constant coefficients—the classical characteristic equation method and the eigenvalue–eigenvector-based matrix solution method—are comparatively examined through a two-compartment pharmacokinetic model. First, the system was reduced to a single second-order differential equation using the classical method, and the solution was obtained through the characteristic roots. Subsequently, the system was represented in matrix form, and the analytical solution was derived through the use of eigenvalues and eigenvectors. A comparison of both approaches revealed that they yield mathematically equivalent solutions. The comparative analysis indicates that the classical method offers a simpler structure in terms of operational steps for two-compartment pharmacokinetic models, whereas the eigenvalue–eigenvector approach provides a generalizable framework for solving higher-dimensional systems. Overall, the study highlights an interdisciplinary connection between theoretical analysis and modeling applications by comparatively evaluating mathematical methods used in pharmacokinetic systems.