In this chapter, we focus our attention on obtaining analytical solutions of linear differential equations with coefficients that are not constant. These solutions are not as simple as those for which the coefficients were constant. One general approach is to use a power series formulation.
In Section 9.1, we describe the main approaches of power series solution. Depending on the equation, one can choose to expand the solution around an ordinary point or a singular point. Each of these choices will determine the structure of the series. For an ordinary point, the expansion is simply a Taylor series, whereas for a singular point, we need a series known as a Frobenius series.
Although the power series method is straightforward, power series solutions can be quite lengthy and complicated. Nonetheless, for certain equations, solutions can be found based on the parameters of the equations, thus yielding direct solutions. This is the case for two important classes of second-order equations that have several applications. These are the Legendre equations and Bessel equations, which we describe in Sections 9.2 and 9.3, respectively.
We have also included other important equations in the exercises, such as hyper-geometric equations, Jacobi equations, Laguerre equations, Hermite equations, and so forth, where the same techniques given in this chapter can be used to generate the useful functions and polynomials. Fortunately, the special functions and polynomials that solve these equations, including Legendre polynomials, Legendre functions and Bessel functions, are included in several computer software programs such as MATLAB.