1. The problem asks if analytic methods can be used to solve the equation $$f(E) = M - Et - e \sin E = 0$$. 2. This is a transcendental equation because it involves both the variable $E$ and the sine function of $E$. 3. Analytic methods typically solve algebraic equations or those reducible to algebraic forms. However, transcendental equations like this often cannot be solved exactly using elementary functions. 4. One can use iterative numerical methods such as Newton-Raphson or fixed-point iteration to approximate solutions. 5. Therefore, analytic closed-form solutions generally do not exist for this equation, and numerical methods are justified. Final answer: Analytic methods cannot generally solve $$M - Et - e \sin E = 0$$ exactly; numerical methods are required.