Second order ordinary differential equations that cannot be solved by analytical methods (as shown in Chapters 50 and 51), i.e. those involving variable coefficients, can often be solved in the form of an infinite series of powers of the variable. This chapter looks at some of the methods that make this possible-by the Leibniz-Maclaurin 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 27 and 32),

(ii) appreciate the binomial theorem (as explained in Chapters 7), and

(iii) use Maclaurins theorem (as explained in Chapter 8).