When encountering differential equations with variable coefficients—such as the famous Bessel or Legendre equations—standard elementary methods often fail to yield solutions in terms of simple functions like sines, cosines, or exponentials. In this chapter, we pivot to a powerful alternative: representing the unknown function as an infinite power series. By using this method, we transform a complex differential equation into a set of algebraic recurrence relations for the coefficients. This approach not only allows us to solve equations that are otherwise ‘unsolvable,’ but it also provides a window into the behavior of functions near both ordinary and singular points, forming the backbone of mathematical physics and engineering analysis.