Page 309 - Schaum's Outline of Differential Equations

292 SOME CLASSICAL DIFFERENTIAL EQUATIONS [CHAP. 29

Solved Problems

29.1. Let n = 2 in the Reunite DE. Use the Rodrigues formula to find the polynomial solution.
The Hermite DE becomes y" - 2xy' + 4y = 0. The Rodrigues formula for the Hermite polynomials, H n(x), is
given by

Letting n = 2, we have H 2(x) = This agrees with our listing above and via direct

2
substitution into the DE, we see that 4x - 2 is indeed a solution.
2
Notes: 1) Any non-zero multiple of 4x - 2 is also a solution. 2) When n = 0 in the Rodrigues formula, the "0-th
Derivative" is defined as the function itself. That is,

2
29.2. Given the Laguerre polynomials L^(x) = —x + 1 and L 2(x) = x -4x + 2, show that these two functions
x
are orthogonal with respect to the weight Junction e~ on the interval (0, °°).
Orthogonality of these polynomials with respect to the given weight function means
x
\(-x + 1) (x 2 - 4x + 2)e~ dx = 0. This integral is indeed zero, as is verified by integration by parts and applying
o
L'Hospital's Rule.

29.3. Using the generating function for the Chebyshev polynomials, T n(x), find T 0(x), T^x), and T 2(x).
The desired generating function is given by

Using long division on the left side of this equation and combing like powers of t yields:

2
Hence, T Q(x) = 1, TI(X) = x, and T 2(x) = 2x - 1, which agrees with our list above. We note that, due to the nature of
the computation, the use of the generating function does not provide an efficient way to actually obtain the
Chebyshev polynomials.

29.4. Let n = 4 in the Legendre DE; verify that P 4(x) = (35x 4 - 30x 2 + 3) is a solution.

2
The DE becomes (1 -x ) y"- 2xy' + 20y = 0. Taking the first and second derivatives of PAx), we obtain
Direct substitution into the DE, followed by collecting like
terms of x,

29.5. The Hermite polynomials, H n(x), satisfy the recurrence relation

Verify this relationship for n = 3.