An extensive analysis of Schwarzschild exterior solution

We pursue a detailed analysis of the Schwarzschild geometry around a spherically symmetric, non-rotating, uncharged source and aim to construct the Schwarzschild metric by considering trigonometric, hyperbolic and logarithmic functions of position and time and solving the Einstein field equation. We investigate whether the Schwarzschild exterior solution is indeed independent of the choice of the nature of the function for the first two metric elements in the general expression for the Schwarzschild metric, and is solely dependent upon the centrally symmetric nature of the geometry taken into account.