Page 311 - Schaum's Outline of Differential Equations

294 SOME CLASSICAL DIFFERENTIAL EQUATIONS [CHAP. 29

Use this formula to obtain P 5(x). Compare this to the results given in Problem 29.6.

29.12. Find Pf(x) by following the procedure given in Problem 29.6.
29.13. Following the procedure in Problem 29.7, show that

29.14. Chebyshev polynomials satisfy the recursion formula

Use this result to obtain T 5(x).

29.15. Legendre polynomials satisfy the condition Show that this is true for P 3(x).

29.16. Laguerre polynomials satisfy the condition Show that this is true for L 2(x).

29.17. Laguerre polynomials also satisfy the equation L' n(x) -nL' n_ l(x) + nL n_ j(jc) = 0. Show that this is true for L 3(x).

1
29.18. Generate HI(X) by using the equation e " ' =

29.19. Consider the "operator" equation , where m, n = 0, 1, 2, 3, .... The polynomials derived from this
equation are called Associated Laguerre polynomials, and are denoted L™(x). Find L%(x) and L\(x).

29.20. Determine whether the five following differential equations have two polynomial solutions; if they do, give the
2
2
2
degrees of the solutions: a) (1 - x )/'+ 5xy'- 5y = 0; b) (1 -x )y"+8xy'- 18y = 0; c) (1 -x )y" + 2xy'+ Wy = 0;
2
2
d)(l- x )y" + 14xy' - 56y = 0; e) (1 - x )y" + 12xy' - 22y = 0.