Page 307 - Schaum's Outline of Differential Equations

CHAPTER 29

Some Classical

Differential Equations

CLASSICAL DIFFERENTIAL EQUATIONS

Because some special differential equations have been studied for many years, both for the aesthetic beauty
of their solutions and because they lend themselves to many physical applications, they may be considered
classical. We have already seen an example of such an equation, the equation of Legendre, in Problem 27.13.
We will touch upon four classical equations: the Chebyshev differential equation, named in honor of Pafnuty
Chebyshey (1821-1894); the Hermite differential equation, so named because of Charles Hermite (1822-1901);
the Laguerre differential equation, labeled after Edmond Laguerre (1834-1886); and the Legendre differential
equation, so titled because of Adrien Legendre (1752-1833). These equations are given in Table 29-1 below:

Table 29-1
(Note: n = 0, 1,2,3, ...)

2
2
Chebyshev Differential Equation (1 - x ) y" -xy' + n y = 0
Hermite Differential Equation y" - 2xy' + 2ny = 0
Laguerre Differential Equation xy" + (1 - *)/ + ny = 0

2
Legendre Differential Equation (1 - x )y" - 2xy' + n(n + l)y =0

POLYNOMIAL SOLUTIONS AND ASSOCIATED CONCEPTS
One of the most important properties these four equations possess, is the fact that they have polynomial
solutions, naturally called Chebyshev polynomials, Hermite polynomials, etc.
There are many ways to obtain these polynomial solutions. One way is to employ series techniques, as
discussed in Chapters 27 and 28. An alternate way is by the use of Rodrigues formulas, so named in honor of
O. Rodrigues (1794-1851), a French banker. This method makes use of repeated differentiations (see, for example,
Problem 29.1).
These polynomial solutions can also be obtained by the use of generating Junctions. In this approach, infinite
series expansions of the specific function "generates" the desired polynomials (see Problem 29.3). It should be
noted, from a computational perspective, that this approach becomes more time-consuming the further along
we go in the series.

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