Optimization problems are utilized to determine the most optimal solution among all possible options, typically while considering certain constraints. When these constraints are associated with physics, it is common to utilize differential equations to describe the phenomenon. The objective of this presentation is to numerically solve the problem of an inverted elastic pendulum. To achieve this goal, we will utilize the indirect method and apply various discretization techniques such as Forward Euler, Backward Euler, and Mid-Point methods to establish a system. Subsequently, we will employ the conjugate gradient, GMRES (Generalized Minimal Residual Method), and monolithic methods to solve the system.