The objective of this study is the development of a mathematical model that represents the tensions and the deflection of a cylindrical pressure vessel under variable internal loads along the length. In order to reach this objective, the study followed three stages: the preparation of an associated physical model, the representation of this model in an algebraic model using numerical methods (finite differences) and the implementation of a programming code for the solution of the algebraic model. In the preparation of the physical model, we considered: equations of balance of forces acting in a control volume defined in cylindrical coordinates; the tensor of the deformations of this volume associated with the tensions induced by an external load and the corresponding linear constitutive relations for this control volume. The finite difference method was used to transform the differential equations resulting from the physical model. In this way, it is possible to represent in the continuous control volume the equations in a discrete mesh in points of the domain, reducing the differential equations to the algebraic form. The implementation of the code to solve the resulting algebraic model is done in the MATLAB program environment, using the available packages for the formation and solution of linear systems. Numerical tests for thick-walled cylinders and thin-walled cylinders are presented at the end of the paper. By comparing the results obtained with the respective analytical solutions, the efficacy and accuracy of the considered model is proven, indicating its robustness for the analysis of more complex problems.