Power series methods of solving ordinary differential equations

Second order ordinary differential equations that cannot be solved by analytical methods (as shown in Chapters 50 and 51), i.e. those involvingvariable 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 27 and 32),

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

(iii) useMaclaurins theorem(as explained inChapter 8).

The following is an extension of successive differentia-