A transcendental function is one that is not a polynomial or a rational function or a root. The higher transcendental functions are ones that are not elementary and are defined using power series. This chapter shows how to solve differential equations using the method of power series, after providing a very brief introduction on how special functions arise from this process. It demonstrates how to use power series to solve first-order linear equations. The solution of ordinary differential equations near singular points of arbitrary type is extremely difficult. Many equations are intractable. Fortunately, many equations that arise in practice, or from physical applications, are of a particularly tame sort. The chapter shows how to reduce the analysis of the heat equation in a three-dimensional example to the study of Legendre’s equation.