on solving a non-linear equation in five-dimensional minimal gauged supergravity

We investigate a class of non-linear differential equations arising from five-dimensional minimal gauged supergravity. specifically, we focus on the structure of the scalar function governing supersymmetric backgrounds and show how its governing equation can be reduced to a solvable non-linear system. by employing an analytic approximation and numerical methods, we present explicit solutions and analyze their physical interpretation within the ads/cft framework. our results highlight the interplay between geometry, gauge fields, and supersymmetry, and extend the known catalogue of exact solutions.