Differential Equations and Linear Algebra, 4th Edition

11.3 Frobenius Series Solutions

We now investigate the solution of the homogeneous second-order linear equation

$$A(x)y″+B(x)y\text{′}+C(x)y=0$$ (1)

near a singular point. Recall that if the functions A, B, and C are polynomials having no common factors, then the singular points of Eq. (1) are simply those points where $A(x)=0.$ For instance, $x=0$ is the only singular point of the Bessel equation of order n,

$$x^{2}y″+xy\text{′}+(x^2-n^2)y=0,$$

whereas the Legendre equation of order n,

$$(1-x^2)y″-2xy\text{′}+n(n+1)y=0,$$

has the two singular points $x=-1$ and $x=1.$ It turns out that some of the features of the solutions of such equations of the most importance for applications are largely ...