Section 11.1

1. $y (x) = c_{0} (1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{3!} + \dots) = c_{0} e^{x}$ $y (x) = c_{0} (1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{3!} + \dots) = c_{0} e^{x}$ ; $ρ = + \infty$ $ρ = + \infty$
2. $y (x) = c_{0} (1 + \frac{4 x}{1!} + \frac{4^{2} x^{2}}{2!} + \frac{4^{3} x^{3}}{3!} + \frac{4^{4} x^{4}}{4!} + \dots) = c_{0} e^{4 x}$ $y (x) = c_{0} (1 + \frac{4 x}{1!} + \frac{4^{2} x^{2}}{2!} + \frac{4^{3} x^{3}}{3!} + \frac{4^{4} x^{4}}{4!} + \dots) = c_{0} e^{4 x}$ ; $ρ = \infty$ $ρ = \infty$
3. $y (x) = c_{0} (1 - \frac{3 x}{2} + \frac{{(3 x)}^{2}}{2! 2^{2}} - \frac{{(3 x)}^{3}}{3! 2^{3}} + \frac{{(3 x)}^{4}}{4! 2^{4}} - \dots) = c_{0} e^{- 3 x / 2}$ $y (x) = c_{0} (1 - \frac{3 x}{2} + \frac{{(3 x)}^{2}}{2! 2^{2}} - \frac{{(3 x)}^{3}}{3! 2^{3}} + \frac{{(3 x)}^{4}}{4! 2^{4}} - \dots) = c_{0} e^{- 3 x / 2}$ ; $ρ = + \infty$ $ρ = + \infty$
4. $y (x) = c_{0} (1 - \frac{x^{2}}{1!} + \frac{x^{4}}{2!} - \frac{x^{6}}{3!} + \dots) = c_{0} e^{- x^{2}}$ $y (x) = c_{0} (1 - \frac{x^{2}}{1!} + \frac{x^{4}}{2!} - \frac{x^{6}}{3!} + \dots) = c_{0} e^{- x^{2}}$ ; $ρ = \infty$ $ρ = \infty$
5. $y (x) = c_{0} (1 + \frac{x^{3}}{3} + \frac{x^{6}}{2! 3^{2}} + \frac{x^{9}}{3! 3^{3}} + \dots) = c_{0} \exp (\frac{1}{3} x^{3})$ $y (x) = c_{0} (1 + \frac{x^{3}}{3} + \frac{x^{6}}{2! 3^{2}} + \frac{x^{9}}{3! 3^{3}} + \dots) = c_{0} \exp (\frac{1}{3} x^{3})$ ; $ρ = + \infty$ $ρ = + \infty$
6. $y (x) = c_{0} (1 + \frac{x}{2} + \frac{x^{2}}{4} + \frac{x^{3}}{8} + \frac{x^{4}}{16} + \dots) = \frac{2 c_{0}}{2 - x}$ $y (x) = c_{0} (1 + \frac{x}{2} + \frac{x^{2}}{4} + \frac{x^{3}}{8} + \frac{x^{4}}{16} + \dots) = \frac{2 c_{0}}{2 - x}$ ; $ρ = 2$ $ρ = 2$
7. $y (x) = c_{0} (1 + 2 x + 4 x^{2} + 8 x^{3} + \dots) = \frac{c_{0}}{1 - 2 x}$ $y (x) = c_{0} (1 + 2 x + 4 x^{2} + 8 x^{3} + \dots) = \frac{c_{0}}{1 - 2 x}$ ; $ρ = \frac{1}{2}$ $ρ = \frac{1}{2}$
8. $y (x) = c_{0} (1 + \frac{x}{2} - \frac{x^{2}}{8} + \frac{x^{3}}{16} - \frac{5 x^{4}}{128} + \dots) = c_{0} \sqrt{1 + x}$ $y (x) = c_{0} (1 + \frac{x}{2} - \frac{x^{2}}{8} + \frac{x^{3}}{16} - \frac{5 x^{4}}{128} + \dots) = c_{0} \sqrt{1 + x}$ ; $ρ = 1$ $ρ = 1$
9. $y (x) = c_{0} (1 + 2 x + 3 x^{2} + 4 x^{3} + \dots) = \frac{c_{0}}{{(1 - x)}^{2}}$ $y (x) = c_{0} (1 + 2 x + 3 x^{2} + 4 x^{3} + \dots) = \frac{c_{0}}{{(1 - x)}^{2}}$ ; $ρ = 1$ $ρ = 1$
10. $y (x) = c_{0} (1 - \frac{3 x}{2} + \frac{3 x^{2}}{8} + \frac{x^{3}}{16} + \frac{3 x^{4}}{128} + \dots) = c_{0} {(1 - x)}^{3 / 2}$ $y (x) = c_{0} (1 - \frac{3 x}{2} + \frac{3 x^{2}}{8} + \frac{x^{3}}{16} + \frac{3 x^{4}}{128} + \dots) = c_{0} {(1 - x)}^{3 / 2}$ ; $ρ = 1$ $ρ = 1$
11. $y (x) = c_{0} (1 + \frac{x^{2}}{2!} + \frac{x^{4}}{4!} + \frac{x^{6}}{6!} + \dots) + c_{1} (x + \frac{x^{3}}{3!} + \frac{x^{5}}{5!} + \frac{x^{7}}{7!} + \dots) = c_{0} \cosh x + c_{1} \sinh x$ $y (x) = c_{0} (1 + \frac{x^{2}}{2!} + \frac{x^{4}}{4!} + \frac{x^{6}}{6!} + \dots) + c_{1} (x + \frac{x^{3}}{3!} + \frac{x^{5}}{5!} + \frac{x^{7}}{7!} + \dots) = c_{0} \cosh x + c_{1} \sinh x$ ; $ρ = + \infty$ $ρ = + \infty$
12. $y (x) = c_{0} (1 + \frac{{(2 x)}^{2}}{2!} + \frac{{(2 x)}^{4}}{4!} + \frac{{(2 x)}^{6}}{6!} + \dots) + \frac{c_{1}}{2} ((2 x) + \frac{{(2 x)}^{3}}{3!} + \frac{{(2 x)}^{5}}{5!} + \frac{{(2 x)}^{7}}{7!} + \dots) = c_{0} \cosh 2 x + \frac{c_{1}}{2} \sinh 2 x$ $y (x) = c_{0} (1 + \frac{{(2 x)}^{2}}{2!} + \frac{{(2 x)}^{4}}{4!} + \frac{{(2 x)}^{6}}{6!} + \dots) + \frac{c_{1}}{2} ((2 x) + \frac{{(2 x)}^{3}}{3!} + \frac{{(2 x)}^{5}}{5!} + \frac{{(2 x)}^{7}}{7!} + \dots) = c_{0} \cosh 2 x + \frac{c_{1}}{2} \sinh 2 x$ ; $ρ = \infty$ $ρ = \infty$
13. $y (x) = c_{0} (1 - \frac{{(3 x)}^{2}}{2!} + \frac{{(3 x)}^{4}}{4!} - \frac{{(3 x)}^{6}}{6!} + \dots) + \frac{c_{1}}{3} (3 x - \frac{{(3 x)}^{3}}{3!} + \frac{{(3 x)}^{5}}{5!} - \frac{{(3 x)}^{7}}{7!} + \dots) = c_{0} \cos 3 x + \frac{1}{3} c_{1} \sin 3 x$ $y (x) = c_{0} (1 - \frac{{(3 x)}^{2}}{2!} + \frac{{(3 x)}^{4}}{4!} - \frac{{(3 x)}^{6}}{6!} + \dots) + \frac{c_{1}}{3} (3 x - \frac{{(3 x)}^{3}}{3!} + \frac{{(3 x)}^{5}}{5!} - \frac{{(3 x)}^{7}}{7!} + \dots) = c_{0} \cos 3 x + \frac{1}{3} c_{1} \sin 3 x$ ; $ρ = + \infty$ $ρ = + \infty$
14. $y (x) = x + c_{0} (1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + \dots) + (c_{1} - 1) (x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + \dots) = x + c_{0} \cos x + (c_{1} - 1) \sin x$ $y (x) = x + c_{0} (1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + \dots) + (c_{1} - 1) (x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + \dots) = x + c_{0} \cos x + (c_{1} - 1) \sin x$ ; $ρ = \infty$ $ρ = \infty$
15. $(n + 1) c_{n} = 0$ $(n + 1) c_{n} = 0$ for all $n ≧ 0,$ $n ≧ 0,$ so $c_{n} = 0$ $c_{n} = 0$ for all $n ≧ 0$ $n ≧ 0$ .
16. $2 n c_{n} = c_{n}$ $2 n c_{n} = c_{n}$ for all $n ≧ 0,$ $n ≧ 0,$ so $c_{n} = 0$ $c_{n} = 0$ for all $n ≧ 0$ $n ≧ 0$ .
17. $c_{0} = c_{1} = 0$ $c_{0} = c_{1} = 0$ and $c_{n + 1} = - n c_{n}$ $c_{n + 1} = - n c_{n}$ for $n ≧ 1,$ $n ≧ 1,$ thus $c_{n} = 0$ $c_{n} = 0$ for all $n ≧ 0$ $n ≧ 0$ .
18. $c_{n} = 0$ $c_{n} = 0$ for all $n ≧ 0$ $n ≧ 0$
19. $(n + 1) (n + 2) c_{n + 2} = - 4 c_{n}$ $(n + 1) (n + 2) c_{n + 2} = - 4 c_{n}$ ; $y (x) = \frac{3}{2} [(2 x) - \frac{{(2 x)}^{3}}{3!} + \frac{{(2 x)}^{5}}{5!} - \frac{{(2 x)}^{7}}{7!} + \dots] = \frac{3}{2} \sin 2 x$ $y (x) = \frac{3}{2} [(2 x) - \frac{{(2 x)}^{3}}{3!} + \frac{{(2 x)}^{5}}{5!} - \frac{{(2 x)}^{7}}{7!} + \dots] = \frac{3}{2} \sin 2 x$
20. $(n + 1) (n + 2) c_{n + 2} = 4 c_{n}$ $(n + 1) (n + 2) c_{n + 2} = 4 c_{n}$ ; $y (x) = 2 [1 + \frac{{(2 x)}^{2}}{2!} + \frac{{(2 x)}^{4}}{4!} + \frac{{(2 x)}^{6}}{6!} + \dots] = 2 \cosh 2 x$ $y (x) = 2 [1 + \frac{{(2 x)}^{2}}{2!} + \frac{{(2 x)}^{4}}{4!} + \frac{{(2 x)}^{6}}{6!} + \dots] = 2 \cosh 2 x$
21. $n (n + 1) c_{n + 1} = 2 n c_{n} - c_{n - 1}$ $n (n + 1) c_{n + 1} = 2 n c_{n} - c_{n - 1}$ ; $y (x) = x + x^{2} + \frac{x^{3}}{2!} + \frac{x^{4}}{3!} + \frac{x^{5}}{4!} + \dots = x e^{x}$ $y (x) = x + x^{2} + \frac{x^{3}}{2!} + \frac{x^{4}}{3!} + \frac{x^{5}}{4!} + \dots = x e^{x}$
22. $n (n + 1) c_{n + 1} = - {n c}_{n} + 2 c_{n - 1}$ $n (n + 1) c_{n + 1} = - {n c}_{n} + 2 c_{n - 1}$ ; $y = e^{- 2 x}$ $y = e^{- 2 x}$
23. As $c_{0} = c_{1} = 0$ $c_{0} = c_{1} = 0$ and $(n^{2} - n + 1) c_{n} + (n - 1) c_{n - 1} = 0$ $(n^{2} - n + 1) c_{n} + (n - 1) c_{n - 1} = 0$ for $n ≧ 2, c_{n} = 0$ $n ≧ 2, c_{n} = 0$ for all $n ≧ 0$ $n ≧ 0$

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Table of Contents for Section 11.1

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Table of Contents for
Section 11.1