Series Solutions of ODEs. Special Functions
In the previous chapters, we have seen that linear ODEs with constant coefficients can be solved by algebraic methods, and that their solutions are elementary functions known from calculus. For ODEs with variable coefficients the situation is more complicated, and their solutions may be nonelementary functions. Legendre's, Bessel's, and the hypergeometric equations are important ODEs of this kind. Since these ODEs and their solutions, the Legendre polynomials, Bessel functions, and hypergeometric functions, play an important role in engineering modeling, we shall consider the two standard methods for solving such ODEs.
The first method is called the power series method because it gives solutions in the form of a power series a0 + a1x + a2x2 + a3x3 + ….
The second method is called the Frobenius method and generalizes the first; it gives solutions in power series, multiplied by a logarithmic term or a fractional power xr, in cases such as Bessel's equation, in which the first method is not general enough.
All those more advanced solutions and various other functions not appearing in calculus are known as higher functions or special functions, which has become a technical term. Each of these functions is important enough to give it a name and investigate its properties and relations to other functions in great detail (take a look ...