The paper offers an approach to the direct finding of traveling wave solutions for some nonlinear partial differential equations with important applications in physics. The basic tool is the auxiliary equation, an equation whose known solutions are used for expressing solutions of more complicated models. The main question we are looking for an answer is how the solutions depend on the auxiliary equations, and the conclusion is that the dependency is related both to the choice of the auxiliary equation and to the solving method. To effectively illustrate this assertion, we will investigate two different models through two distinct methods. The considered models are Dodd-Bullough-Mikhailov and the Benjamin-Bona-Mahony equations. Each of them will be solved using two different auxiliary equations, and will generate interesting classes of rational, hyperbolic or periodic solutions.