Power series methods of solving ordinary differential equations

At the end of this chapter, you should be able to:

• appreciate the reason for using power series methods to solve differential equations • determine higher-order differential coefficients as a series • use Leibniz’s theorem to obtain the nth derivative of a given function • obtain a power series solution of a differential equation by the Leibniz-Maclaurin method • obtain a power series solution of a differential equation by the Frobenius method • determine the general power series solution of Bessel’s equation • express Bessel’s equation in terms of gamma functions • determine the general power series solution of Legendre’s equation • determine Legendre polynomials • determine Legendre polynomials using Rodrigue’s formula

Second-order ordinary differential equations that cannot be solved by analytical methods (as shown in Chapters 81 and 82), i.e. those involving variable coefficients, can often be solved in the formof an infinite series of powers of the variable. This chapter looks at some of the methods that make this possible – by the LeibnizMaclaurin and Frobinius methods, involving Bessel’s and Legendre’s equations, Bessel and gamma functions and Legendre’s polynomials. Before introducing Leibniz’s theorem, some trends with higher differential coefficients are considered. To better understand this chapter it is necessary to be able to:

(i) differentiate standard functions (as explained in Chapters 53 and 58),

(ii) appreciate the binomial theorem (as explained in Chapter 22), and

(iii) use Maclaurin’s theorem (as explained in Chapter 23).