In solving the pulsar equation, two methods have risen to the forefront, the CKF method (Contopoulos, Kazanas, and Fendt), and the TOTS method (Takamori, Okawa, Takamoto, and Suwa). Both methods are implemented by numerical relaxation, which creates problems at a singular surface known as the light cylinder. Furthermore, these methods give limited information about the problem. The CKF method will not tell you how singular an answer is, only what it will look like after iterative correction. The TOTS method, which foregoes iterative correction, has the potential to give more information, but it has only been tested once, and an extra physical quantity was unnecessarily restricted just to get the solution to converge. We have replaced relaxation with Newton's method in the context of solving nonlinear equations. This technique is demonstrated by replicating the results of Michel 1973, Contopoulos et al. 1999, Takamori et al. 2012, Lovelace et al. 2006, and Contopoulos et al. 2014. These altered methods refine the original ideas and make clear exactly what place they have in searching for solutions. We also introduce new investigative paths. We show how we can weed out solutions by revealing the singular behavior of a derivative, even if the function itself appears well-behaved. We also show how we can use a singular solution to generate a smooth solution by looking for smooth contour lines in a field of singular ones with the "lone contour method."