  This work aims to obtain a positive, smooth, even, and homoclinic to zero
(i.e zero at infinity) solution to a non-autonomous, second-order, nonlinear
differential equation involving quadratic growth on the derivative. We apply
Galerkin's method combined with Strauss' approximation on the term involving
the first derivative to obtain weak solutions. We also study the regularity of
the solutions and the dependence on their existence with a parameter
