13 Second-order
differential equations.
Some special functions
13.1 Concepts
We saw in Sections 12.5 to 12.7 how three very different physical problems are
modelled in terms of the samedifferential equation: equation (12.35) for the classical
harmonic oscillator as an initial value problem, the same equation, (12.47), for the
quantum-mechanical particle in a box as a boundary value problem, and (12.62) for
the quantum-mechanical particle in a ring as a periodic boundary value problem.
This is a common phenomenon in the mathematical modelling of physical systems,
and a number of equations are important enough to have been given names. Some of
these are listed in Table 13.1.
Table 13.1
Name Equation
Legendre (1 1 − 1 x
2
)y′′ 1 − 12 xy′ 1 + 1 l(l 1 + 1 1)y 1 = 10
Associated Legendre (1 1 − 1 x
2
)y′′ 1 − 12 xy′ 1 + 1 [l(l 1 + 1 1) 1 − 1 m
2
2 (1 1 − 1 x
2
)]y 1 = 10
Hermite y′′ 1 − 12 xy′ 1 + 12 ny 1 = 10
Laguerre xy′′ 1 + 1 (1 1 − 1 x)y′ 1 + 1 ny 1 = 10
Associated Laguerre xy′′ 1 + 1 (m 1 + 111 − 1 x)y′ 1 + 1 (n 1 − 1 m)y 1 = 10
Bessel x
2
y′′ 1 + 1 xy′ 1 + 1 (x
2
1 − 1 n
2
)y 1 = 10
The equations in Table 13.1 are all second-order homogeneous linear equations with
non-constant coefficients, and they have particular solutions that play an important
role in the mathematics of the physical sciences. These solutions are often called
special functions.
The standard methods used to solve the equations in Table 13.1 and many other
linear differential equations in the sciences are the power-series method, described
in Section 13.2, and the more general Frobenius method, outlined in Section 13.3.
The former is used in Section 13.4 to obtain the physically-significant particular
solutions of the Legendre equation. These solutions are called Legendre polynomials
and they occur whenever a physical problem in three dimensions is formulated
in terms of spherical polar coordinates. Subsequent sections are devoted to brief
descriptions of the other special functions that are solutions of the equations listed
in Table 13.1.