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Some differential equations may have polynomial solutions.
This section discusses how to obtain polynomial solutions to some classes of differential equations.
The Chebyshev polynomials were discovered in 1854 by the Russian mathematician Pafnuty Chebyshev (1821--1894) in his paper Théorie des mécanismes connus sous le nom de parallélogrammes.
Rather than viewing applications as the beneficiary of elegant mathematics, Chebyshev had a broader approach to describe the nature with mathematical tools. idea. In fact, he found many of his greatest theoretical mathematical discoveries by observing mechanical systems (click here to see the pictures).
His discovery of Chebyshev polynomials was motivated by the mechanical problem how to convert a rotation into horizontal movement observed in a steam engine of train (nineteen century). Pafnuty also invented other polynomials now known as Hermite and Laguerre.
For a more extensive account of the history of this discovery, see The theory of best approximation of functions.
By the way, Chebyshev used seven different spellings of his last name in Latin letters (including French transliteration Tchebycheff or Tchebychev of his name; that is why the letter ''T'' is used for his polynomials of the first kind) and two Russian spellings.
which was fist discovered by
the Russian mathematician Pafnuty Chebyshev (1821--1894) in 1859. We solve this equation, called the Chebyshev--Laguerre equation, by power series method:
Although solutions of the Laguerre equation and the Chebyshev--Laguerre equation are expressed through similar formulas, their behaviors are completely different.
This section is devoted to some second order differential
equations that admit polynomial solutions. To motivate the reader, let us
consider a generalized Hermite equation
\[
y'' -p\,x^M y' + p\,r\,x^{M-1} y =0 ,
\]
where p ≠ 0 and M and r are positive integers. The
above equation becomes standard Hermite equation for M=1, p=2,
and r a positive integer that admits the well-known Hermite polynomial
solution.
Theorem:
The equation \( y'' -p\,x^M y' + p\,r\,x^{M-1} y =0 \) has a polynomial solution of degree r if and only if
\(
r = k\left( M+1 \right) \)
or
\( r = k\left( M+1 \right) +1 \)
for some k = 0,1,2,.... ⦻⧫
Formal substitution of \( y(x) = \sum_{n\ge 0} a_n x^n \) into the given equation gives the recursive formula:
Example:
Consider the differential equation \( y'' - 3\,x^5 y' + 39\,x^4 y =0 , \) which corresponds to p=3, M=5, and
r=7.
According to Theorem, this differential equation admits a polynomial solution.
Substituting the series \( y = \sum_{n\ge 0} a_n x^n \) into the given equation, we obtain
Example:
Consider the differential equation \( y'' - x^2 y' +
5\,x\, y =0 , \) which corresponds to p=3, M=5, and
r=7.
According to Theorem, this differential equation does not have a polynomial
solution because there is no such integer k that satisfies either
5 = k(2+1) = 3k or 5 = k(2+1) +1 = 3k+1.
■
Nonhomogeneous Hermite equations
Now we turn our attention to nonhomogeneous equations.
Theorem:
The initial value problem for the nonhomogeneous Hermite equation
\[
b = -(2n+1) \prod_{i=0}^n \frac{1}{4i+2-4m} .
\]
⧫
Costa, G.B. and Levine, L.E., Polynomial solutions of cirtain classes of ordinary differential equations, Intenational Journal of Mathematical Education in Science and Technology, 1989, Vol. 20, No 1, pp. 1--11; doi: https://doi.org/10.1080/0020739890200101
Levine, L.E. and Maleh, R., Polynomial solutions of the classical equations of Hermite, Legendre, and Chebyshev, Intenational Journal of Mathematical Education in Science and Technology, 2003, Vol. 34, No 1, pp. 95--103; doi: https://doi.org/10.1080/0020739021000053891
Sezer M., Kaynak, M., Chebyshev polynomial solutions of linear differential equations, Intenational Journal of Mathematical Education in Science and Technology, 1996, Vol. 27, No. 4, pp. 607--618.
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