Differential Equations and Linear Algebra, 4th Edition

11.2 Power Series Solutions

The power series method introduced in Section 11.1 can be applied to linear equations of any order (as well as to certain nonlinear equations), but its most important applications are to homogeneous second-order linear differential equations of the form

$$A(x)y″+B(x)y\text{′}+C(x)y=0,$$ (1)

where the coefficients A, B, and C are analytic functions of x. Indeed, in most applications these coefficient functions are simple polynomials.

We saw in Example 3 of Section 11.1 that the series method does not always yield a series solution. To discover when it does succeed, we rewrite Eq. (1) in the form

$$y″+P(x)y\text{′}+Q(x)y=0$$ (2)

with leading ...