Section 11.2

1. $c_{n + 2} = c_{n}$ $c_{n + 2} = c_{n}$ ; $y (x) = c_{0} \sum_{n = 0}^{\infty} x^{2 n} + c_{1} \sum_{n = 0}^{\infty} x^{2 n + 1} = \frac{c_{0} + c_{1} x}{1 - x^{2}}$ $y (x) = c_{0} \sum_{n = 0}^{\infty} x^{2 n} + c_{1} \sum_{n = 0}^{\infty} x^{2 n + 1} = \frac{c_{0} + c_{1} x}{1 - x^{2}}$ ; $ρ = 1$ $ρ = 1$
2. $c_{n + 2} = - \frac{1}{2} c_{n}$ $c_{n + 2} = - \frac{1}{2} c_{n}$ ; $ρ = 2$ $ρ = 2$ ; $y (x) = c_{0} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{2 n}}{2^{n}} + c_{1} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{2 n + 1}}{2^{n}}$ $y (x) = c_{0} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{2 n}}{2^{n}} + c_{1} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{2 n + 1}}{2^{n}}$
3. $(n + 2) c_{n + 2} = - c_{n}$ $(n + 2) c_{n + 2} = - c_{n}$ ; $y (x) = c_{0} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{2 n}}{n! 2^{n}} + c_{1} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{2 n + 1}}{(2 n + 1)!!}$ $y (x) = c_{0} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{2 n}}{n! 2^{n}} + c_{1} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{2 n + 1}}{(2 n + 1)!!}$ ; $ρ = + \infty$ $ρ = + \infty$
4. $(n + 2) c_{n + 2} = - (n + 4) c_{n}$ $(n + 2) c_{n + 2} = - (n + 4) c_{n}$ ; $ρ = 1$ $ρ = 1$ ; $c_{0} \sum_{n = 0}^{\infty} {(- 1)}^{n} (n + 1) x^{2 n} + \frac{1}{3} c_{1} \sum_{n = 0}^{\infty} {(- 1)}^{n} (2 n + 3) x^{2 n + 1}$ $c_{0} \sum_{n = 0}^{\infty} {(- 1)}^{n} (n + 1) x^{2 n} + \frac{1}{3} c_{1} \sum_{n = 0}^{\infty} {(- 1)}^{n} (2 n + 3) x^{2 n + 1}$
5. $3 (n + 2) c_{n + 2} = n c_{n}$ $3 (n + 2) c_{n + 2} = n c_{n}$ ; $ρ = \sqrt{3}$ $ρ = \sqrt{3}$ ; $y (x) = c_{0} + c_{1} \sum_{n = 0}^{\infty} \frac{x^{2 n + 1}}{(2 n + 1) 3^{n}}$ $y (x) = c_{0} + c_{1} \sum_{n = 0}^{\infty} \frac{x^{2 n + 1}}{(2 n + 1) 3^{n}}$
6. $(n + 1) (n + 2) c_{n + 2} = (n - 3) (n - 4) c_{n}$ $(n + 1) (n + 2) c_{n + 2} = (n - 3) (n - 4) c_{n}$ ; $ρ = \infty$ $ρ = \infty$ ; $y (x) = c_{0} (1 + 6 x^{2} + x^{4}) + c_{1} (x + x^{3})$ $y (x) = c_{0} (1 + 6 x^{2} + x^{4}) + c_{1} (x + x^{3})$
7. $3 (n + 1) (n + 2) c_{n + 2} = - {(n - 4)}^{2} c_{n}$ $3 (n + 1) (n + 2) c_{n + 2} = - {(n - 4)}^{2} c_{n}$ ; $y (x) = c_{0} (1 - \frac{8 x^{2}}{3} + \frac{8 x^{4}}{27}) + c_{1} (x - \frac{x^{3}}{2} + \frac{x^{5}}{120} + 9 \sum_{n = 3}^{\infty} \frac{{(- 1)}^{n} {[(2 n - 5)!!]}^{2} x^{2 n + 1}}{(2 n + 1)! 3^{n}})$ $y (x) = c_{0} (1 - \frac{8 x^{2}}{3} + \frac{8 x^{4}}{27}) + c_{1} (x - \frac{x^{3}}{2} + \frac{x^{5}}{120} + 9 \sum_{n = 3}^{\infty} \frac{{(- 1)}^{n} {[(2 n - 5)!!]}^{2} x^{2 n + 1}}{(2 n + 1)! 3^{n}})$
8. $2 (n + 1) (n + 2) c_{n + 2} = (n - 4) (n + 4) c_{n}$ $2 (n + 1) (n + 2) c_{n + 2} = (n - 4) (n + 4) c_{n}$ ; $y (x) = c_{0} (1 - 4 x^{2} + 2 x^{4}) + c_{1} (x - \frac{5 x^{3}}{4} + \frac{7 x^{5}}{32} + \sum_{n = 3}^{\infty} \frac{(2 n - 5)!! (2 n + 3)!! x^{2 n + 1}}{(2 n + 1)! 2^{n}})$ $y (x) = c_{0} (1 - 4 x^{2} + 2 x^{4}) + c_{1} (x - \frac{5 x^{3}}{4} + \frac{7 x^{5}}{32} + \sum_{n = 3}^{\infty} \frac{(2 n - 5)!! (2 n + 3)!! x^{2 n + 1}}{(2 n + 1)! 2^{n}})$
9. $(n + 1) (n + 2) c_{n + 2} = (n + 3) (n + 4) c_{n}$ $(n + 1) (n + 2) c_{n + 2} = (n + 3) (n + 4) c_{n}$ ; $ρ = 1$ $ρ = 1$ ; $y (x) = c_{0} \sum_{n = 0}^{\infty} (n + 1) (2 n + 1) x^{2 n} + \frac{c_{1}}{3} \sum_{n = 0}^{\infty} (n + 1) (2 n + 3) x^{2 n + 1}$ $y (x) = c_{0} \sum_{n = 0}^{\infty} (n + 1) (2 n + 1) x^{2 n} + \frac{c_{1}}{3} \sum_{n = 0}^{\infty} (n + 1) (2 n + 3) x^{2 n + 1}$
10. $3 (n + 1) (n + 2) c_{n + 2} = - (n - 4) c_{n}$ $3 (n + 1) (n + 2) c_{n + 2} = - (n - 4) c_{n}$ ; $y (x) = c_{0} (1 + \frac{2 x^{2}}{3} + \frac{x^{4}}{27}) + c_{1} (x + \frac{x^{3}}{6} + \frac{x^{5}}{360} + 3 \sum_{n = 3}^{\infty} \frac{{(- 1)}^{n} (2 n - 5)!! x^{2 n + 1}}{(2 n + 1)! 3^{n}})$ $y (x) = c_{0} (1 + \frac{2 x^{2}}{3} + \frac{x^{4}}{27}) + c_{1} (x + \frac{x^{3}}{6} + \frac{x^{5}}{360} + 3 \sum_{n = 3}^{\infty} \frac{{(- 1)}^{n} (2 n - 5)!! x^{2 n + 1}}{(2 n + 1)! 3^{n}})$
11. $5 (n + 1) (n + 2) c_{n + 2} = 2 (n - 5) c_{n}$ $5 (n + 1) (n + 2) c_{n + 2} = 2 (n - 5) c_{n}$ ; $y (x) = c_{1} (x - \frac{4 x^{3}}{15} + \frac{4 x^{5}}{375}) + c_{0} (1 - x^{2} + \frac{x^{4}}{10} + \frac{x^{6}}{750} + 15 \sum_{n = 4}^{\infty} \frac{(2 n - 7)!! 2^{n} x^{2 n}}{(2 n)! 5^{n}})$ $y (x) = c_{1} (x - \frac{4 x^{3}}{15} + \frac{4 x^{5}}{375}) + c_{0} (1 - x^{2} + \frac{x^{4}}{10} + \frac{x^{6}}{750} + 15 \sum_{n = 4}^{\infty} \frac{(2 n - 7)!! 2^{n} x^{2 n}}{(2 n)! 5^{n}})$
12. $c_{2} = 0$ $c_{2} = 0$ ; $(n + 2) c_{n + 3} = c_{n}$ $(n + 2) c_{n + 3} = c_{n}$ ; $y (x) = c_{0} (1 + \sum_{n = 1}^{\infty} \frac{x^{3 n}}{2 \cdot 5 \dots (3 n - 1)}) + c_{1} \sum_{n = 0}^{\infty} \frac{x^{3 n + 1}}{n! 3^{n}}$ $y (x) = c_{0} (1 + \sum_{n = 1}^{\infty} \frac{x^{3 n}}{2 \cdot 5 \dots (3 n - 1)}) + c_{1} \sum_{n = 0}^{\infty} \frac{x^{3 n + 1}}{n! 3^{n}}$
13. $c_{2} = 0$ $c_{2} = 0$ ; $(n + 3) c_{n + 3} = - c_{n}$ $(n + 3) c_{n + 3} = - c_{n}$ ; $y (x) = c_{0} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{3 n}}{n! 3^{n}} + c_{1} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{3 n + 1}}{1 \cdot 4 \dots (3 n + 1)}$ $y (x) = c_{0} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{3 n}}{n! 3^{n}} + c_{1} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{3 n + 1}}{1 \cdot 4 \dots (3 n + 1)}$
14. $c_{2} = 0$ $c_{2} = 0$ ; $(n + 2) (n + 3) c_{n + 3} = - c_{n}$ $(n + 2) (n + 3) c_{n + 3} = - c_{n}$ ; $y (x) = c_{0} (1 + \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} x^{3 n}}{3^{n} \cdot n! \cdot 2 \cdot 5 \dots (3 n - 1)}) + c_{1} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{3 n + 1}}{3^{n} \cdot n! \cdot 1 \cdot 4 \dots (3 n + 1)}$ $y (x) = c_{0} (1 + \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} x^{3 n}}{3^{n} \cdot n! \cdot 2 \cdot 5 \dots (3 n - 1)}) + c_{1} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{3 n + 1}}{3^{n} \cdot n! \cdot 1 \cdot 4 \dots (3 n + 1)}$
15. $c_{2} = c_{3} = 0$ $c_{2} = c_{3} = 0$ ; $(n + 3) (n + 4) c_{n + 4} = - c_{n}$ $(n + 3) (n + 4) c_{n + 4} = - c_{n}$ ; $y (x) = c_{0} (1 + \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} x^{4 n}}{4^{n} \cdot n! \cdot 3 \cdot 7 \dots (4 n - 1)}) + c_{1} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{4 n + 1}}{4^{n} \cdot n! \cdot 5 \cdot 9 \dots (4 n + 1)}$ $y (x) = c_{0} (1 + \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} x^{4 n}}{4^{n} \cdot n! \cdot 3 \cdot 7 \dots (4 n - 1)}) + c_{1} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{4 n + 1}}{4^{n} \cdot n! \cdot 5 \cdot 9 \dots (4 n + 1)}$
16. $y (x) = x$ $y (x) = x$
17. $y (x) = 1 + x^{2}$ $y (x) = 1 + x^{2}$
18. $y (x) = 2 \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} {(x - 1)}^{2 n}}{n! 2^{n}}$ $y (x) = 2 \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} {(x - 1)}^{2 n}}{n! 2^{n}}$ ; converges for all x
19. $y (x) = \frac{1}{3} \sum_{n = 0}^{\infty} (2 n + 3) {(x - 1)}^{2 n + 1}$ $y (x) = \frac{1}{3} \sum_{n = 0}^{\infty} (2 n + 3) {(x - 1)}^{2 n + 1}$ ; converges if $0 < x < 2$ $0 < x < 2$
20. $y (x) = 2 - 6 {(x - 3)}^{2}$ $y (x) = 2 - 6 {(x - 3)}^{2}$ ; converges for all x
21. $y (x) = 1 + 4 {(x + 2)}^{2}$ $y (x) = 1 + 4 {(x + 2)}^{2}$ ; converges for all x
22. $y (x) = 2 x + 6$ $y (x) = 2 x + 6$
23. $2 c_{2} + c_{0} = 0$ $2 c_{2} + c_{0} = 0$ ; $(n + 1) (n + 2) c_{n + 2} + c_{n} + c_{n - 1} = 0$ $(n + 1) (n + 2) c_{n + 2} + c_{n} + c_{n - 1} = 0$ for $n ≧ 1$ $n ≧ 1$ ; $y_{1} (x) = 1 - \frac{x^{2}}{2} - \frac{x^{3}}{6} + \dots$ $y_{1} (x) = 1 - \frac{x^{2}}{2} - \frac{x^{3}}{6} + \dots$ ; $y_{2} (x) = x - \frac{x^{3}}{6} - \frac{x^{4}}{12} + \dots$ $y_{2} (x) = x - \frac{x^{3}}{6} - \frac{x^{4}}{12} + \dots$
24. $y_{1} (x) = 1 + \frac{x^{3}}{3} + \frac{x^{5}}{5} + \frac{x^{6}}{45} + \dots$ $y_{1} (x) = 1 + \frac{x^{3}}{3} + \frac{x^{5}}{5} + \frac{x^{6}}{45} + \dots$ ; $y_{2} (x) = x + \frac{x^{3}}{3} + \frac{x^{4}}{6} + \frac{x^{5}}{5} + \dots$ $y_{2} (x) = x + \frac{x^{3}}{3} + \frac{x^{4}}{6} + \frac{x^{5}}{5} + \dots$
25. $c_{2} = c_{3} = 0, (n + 3) (n + 4) c_{n + 4} + (n + 1) c_{n + 1} + c_{n} = 0$ $c_{2} = c_{3} = 0, (n + 3) (n + 4) c_{n + 4} + (n + 1) c_{n + 1} + c_{n} = 0$ for $n ≧ 0$ $n ≧ 0$ ; $y_{1} (x) = 1 - \frac{x^{4}}{12} + \frac{x^{7}}{126} + \dots$ $y_{1} (x) = 1 - \frac{x^{4}}{12} + \frac{x^{7}}{126} + \dots$ ; $y_{2} (x) = x - \frac{x^{4}}{12} - \frac{x^{5}}{20} + \dots$ $y_{2} (x) = x - \frac{x^{4}}{12} - \frac{x^{5}}{20} + \dots$
26. $y (x) = c_{0} (1 - \frac{x^{6}}{30} + \frac{x^{9}}{72} + \dots) + c_{1} (x - \frac{x^{7}}{42} + \frac{x^{10}}{90} + \dots)$ $y (x) = c_{0} (1 - \frac{x^{6}}{30} + \frac{x^{9}}{72} + \dots) + c_{1} (x - \frac{x^{7}}{42} + \frac{x^{10}}{90} + \dots)$
27. $y (x) = 1 - x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{24} + \frac{x^{5}}{30} + \frac{29 x^{6}}{720} + \frac{13 x^{7}}{630} - \frac{143 x^{8}}{40320} + \dots$ $y (x) = 1 - x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{24} + \frac{x^{5}}{30} + \frac{29 x^{6}}{720} + \frac{13 x^{7}}{630} - \frac{143 x^{8}}{40320} + \dots$ ; $y (0.5) \approx 0.4156$ $y (0.5) \approx 0.4156$
28. $y (x) = c_{0} (1 - \frac{x^{2}}{2} + \frac{x^{3}}{6} + \dots) + c_{1} (x - \frac{x^{3}}{6} + \frac{x^{4}}{12} + \dots)$ $y (x) = c_{0} (1 - \frac{x^{2}}{2} + \frac{x^{3}}{6} + \dots) + c_{1} (x - \frac{x^{3}}{6} + \frac{x^{4}}{12} + \dots)$
29. $y_{1} (x) = 1 - \frac{1}{2} x^{2} + \frac{1}{720} x^{6} + \dots$ $y_{1} (x) = 1 - \frac{1}{2} x^{2} + \frac{1}{720} x^{6} + \dots$ ; $y_{2} (x) = x - \frac{1}{6} x^{3} - \frac{1}{60} x^{5} + \dots$ $y_{2} (x) = x - \frac{1}{6} x^{3} - \frac{1}{60} x^{5} + \dots$
30. $y (x) = c_{0} (1 - \frac{x^{2}}{2} + \frac{x^{3}}{6} + \dots) + c_{1} (x - \frac{x^{2}}{2} + \frac{x^{4}}{18} + \dots)$ $y (x) = c_{0} (1 - \frac{x^{2}}{2} + \frac{x^{3}}{6} + \dots) + c_{1} (x - \frac{x^{2}}{2} + \frac{x^{4}}{18} + \dots)$
33. The following figure shows the interlaced zeros of the 4th and 5th Hermite polynomials.
34. The figure below results when we use $n = 40$ $n = 40$ terms in each summation. But with $n = 50$ $n = 50$ we get the same picture as Fig. 8.2.3 in the text.

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Table of Contents for Section 11.2

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Table of Contents for
Section 11.2