Section 11.3

1. Ordinary point
2. Ordinary point
3. Irregular singular point
4. Irregular singular point
5. Regular singular point; $r_{1} = 0, r_{2} = - 1$ $r_{1} = 0, r_{2} = - 1$
6. Regular singular point; $r_{1} = 1, r_{2} = - 2$ $r_{1} = 1, r_{2} = - 2$
7. Regular singular point; $r = - 3, - 3$ $r = - 3, - 3$
8. Regular singular point; $r = \frac{1}{2}, - 3$ $r = \frac{1}{2}, - 3$
9. Regular singular point $x = 1$ $x = 1$
10. Regular singular point $x = 1$ $x = 1$
11. Regular singular points $x = 1, - 1$ $x = 1, - 1$
12. Irregular singular point $x = 2$ $x = 2$
13. Regular singular points $x = 2, - 2$ $x = 2, - 2$
14. Irregular singular points $x = 3, - 3$ $x = 3, - 3$
15. Regular singular point $x = 2$ $x = 2$
16. Irregular singular point $x = 0,$ $x = 0,$ regular singular point $x = 1$ $x = 1$
17. $y_{1} (x) = \cos \sqrt{x}, y_{2} (x) = \sin \sqrt{x}$ $y_{1} (x) = \cos \sqrt{x}, y_{2} (x) = \sin \sqrt{x}$
18. $y_{1} (x) = \sum_{n = 0}^{\infty} \frac{x^{n}}{n! (2 n + 1)!!}, y_{2} (x) = x^{- 1 / 2} \sum_{n = 0}^{\infty} \frac{x^{n}}{n! (2 n - 1)!!}$ $y_{1} (x) = \sum_{n = 0}^{\infty} \frac{x^{n}}{n! (2 n + 1)!!}, y_{2} (x) = x^{- 1 / 2} \sum_{n = 0}^{\infty} \frac{x^{n}}{n! (2 n - 1)!!}$
19. $y_{1} (x) = x^{3 / 2} (1 + 3 \sum_{n = 1}^{\infty} \frac{x^{n}}{n! (2 n + 3)!!}), y_{2} (x) = 1 - x - \sum_{n = 2}^{\infty} \frac{x^{n}}{n! (2 n - 3)!!}$ $y_{1} (x) = x^{3 / 2} (1 + 3 \sum_{n = 1}^{\infty} \frac{x^{n}}{n! (2 n + 3)!!}), y_{2} (x) = 1 - x - \sum_{n = 2}^{\infty} \frac{x^{n}}{n! (2 n - 3)!!}$
20. $y_{1} (x) = x^{1 / 3} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} 2^{n} x^{n}}{n! \cdot 4 \cdot 7 \dots (3 n + 1)}, y_{2} (x) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} 2^{n} x^{n}}{n! \cdot 2 \cdot 5 \dots (3 n - 1)}$ $y_{1} (x) = x^{1 / 3} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} 2^{n} x^{n}}{n! \cdot 4 \cdot 7 \dots (3 n + 1)}, y_{2} (x) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} 2^{n} x^{n}}{n! \cdot 2 \cdot 5 \dots (3 n - 1)}$
21. $y_{1} (x) = x (1 + \sum_{n = 1}^{\infty} \frac{x^{2 n}}{n! \cdot 7 \cdot 11 \dots (4 n + 3)}), y_{2} (x) = x^{- 1 / 2} (1 + \sum_{n = 1}^{\infty} \frac{x^{2 n}}{n! \cdot 1 \cdot 5 \dots (4 n - 3)})$ $y_{1} (x) = x (1 + \sum_{n = 1}^{\infty} \frac{x^{2 n}}{n! \cdot 7 \cdot 11 \dots (4 n + 3)}), y_{2} (x) = x^{- 1 / 2} (1 + \sum_{n = 1}^{\infty} \frac{x^{2 n}}{n! \cdot 1 \cdot 5 \dots (4 n - 3)})$
22. $y_{1} (x) = x^{3 / 2} (1 + \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} x^{2 n}}{n! \cdot 9 \cdot 13 \dots (4 n + 5)}), y_{2} (x) = x^{- 1} (1 + \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n - 1} x^{2 n}}{n! \cdot 3 \cdot 7 \dots (4 n - 1)})$ $y_{1} (x) = x^{3 / 2} (1 + \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} x^{2 n}}{n! \cdot 9 \cdot 13 \dots (4 n + 5)}), y_{2} (x) = x^{- 1} (1 + \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n - 1} x^{2 n}}{n! \cdot 3 \cdot 7 \dots (4 n - 1)})$
23. $y_{1} (x) = x^{1 / 2} (1 + \sum_{n = 1}^{\infty} \frac{x^{2 n}}{2^{n} \cdot n! \cdot 19 \cdot 31 \dots (12 n + 7)}), y_{2} (x) = x^{- 2 / 3} (1 + \sum_{n = 1}^{\infty} \frac{x^{2 n}}{2^{n} \cdot n! \cdot 5 \cdot 17 \dots (12 n - 7)})$ $y_{1} (x) = x^{1 / 2} (1 + \sum_{n = 1}^{\infty} \frac{x^{2 n}}{2^{n} \cdot n! \cdot 19 \cdot 31 \dots (12 n + 7)}), y_{2} (x) = x^{- 2 / 3} (1 + \sum_{n = 1}^{\infty} \frac{x^{2 n}}{2^{n} \cdot n! \cdot 5 \cdot 17 \dots (12 n - 7)})$
24. $y_{1} (x) = x^{1 / 3} (1 + \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} x^{2 n}}{2^{n} \cdot n! \cdot 7 \cdot 13 \dots (6 n + 1)}), y_{2} (x) = 1 + \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} x^{2 n}}{2^{n} \cdot n! \cdot 5 \cdot 11 \dots (6 n + 1)}$ $y_{1} (x) = x^{1 / 3} (1 + \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} x^{2 n}}{2^{n} \cdot n! \cdot 7 \cdot 13 \dots (6 n + 1)}), y_{2} (x) = 1 + \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} x^{2 n}}{2^{n} \cdot n! \cdot 5 \cdot 11 \dots (6 n + 1)}$
25. $y_{1} (x) = x^{1 / 2} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{n}}{n! \cdot 2^{n}} = x^{1 / 2} e^{- x / 2}, y_{2} (x) = 1 + \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} x^{n}}{(2 n - 1)!!}$ $y_{1} (x) = x^{1 / 2} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{n}}{n! \cdot 2^{n}} = x^{1 / 2} e^{- x / 2}, y_{2} (x) = 1 + \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} x^{n}}{(2 n - 1)!!}$
26. $y_{1} (x) = x^{1 / 2} \sum_{n = 0}^{\infty} \frac{x^{2 n}}{n! \cdot 2^{n}} = x^{1 / 2} \exp (\frac{1}{2} x^{2}), y_{2} (x) = 1 + \sum_{n = 1}^{\infty} \frac{2^{n} x^{2 n}}{3 \cdot 7 \dots (4 n - 1)}$ $y_{1} (x) = x^{1 / 2} \sum_{n = 0}^{\infty} \frac{x^{2 n}}{n! \cdot 2^{n}} = x^{1 / 2} \exp (\frac{1}{2} x^{2}), y_{2} (x) = 1 + \sum_{n = 1}^{\infty} \frac{2^{n} x^{2 n}}{3 \cdot 7 \dots (4 n - 1)}$
27. $y_{1} (x) = \frac{1}{x} \cos 3 x, y_{2} (x) = \frac{1}{x} \sin 3 x$ $y_{1} (x) = \frac{1}{x} \cos 3 x, y_{2} (x) = \frac{1}{x} \sin 3 x$
28. $y_{1} (x) = \frac{1}{x} \cosh 2 x, y_{2} (x) = \frac{1}{x} \sinh 2 x$ $y_{1} (x) = \frac{1}{x} \cosh 2 x, y_{2} (x) = \frac{1}{x} \sinh 2 x$
29. $y_{1} (x) = \frac{1}{x} \cos \frac{x}{2}, y_{2} (x) = \frac{1}{x} \sin \frac{x}{2}$ $y_{1} (x) = \frac{1}{x} \cos \frac{x}{2}, y_{2} (x) = \frac{1}{x} \sin \frac{x}{2}$
30. $y_{1} (x) = \cos x^{2}, y_{2} (x) = \sin x^{2}$ $y_{1} (x) = \cos x^{2}, y_{2} (x) = \sin x^{2}$
31. $y_{1} (x) = x^{1 / 2} \cosh x, y_{2} (x) = x^{1 / 2} \sinh x$ $y_{1} (x) = x^{1 / 2} \cosh x, y_{2} (x) = x^{1 / 2} \sinh x$
32. $y_{1} (x) = x + \frac{x^{2}}{5}, y_{2} (x) = x^{- 1 / 2} (1 - \frac{5 x}{2} - \frac{15 x^{2}}{8} - \frac{5 x^{3}}{48} + \dots)$ $y_{1} (x) = x + \frac{x^{2}}{5}, y_{2} (x) = x^{- 1 / 2} (1 - \frac{5 x}{2} - \frac{15 x^{2}}{8} - \frac{5 x^{3}}{48} + \dots)$
33. $y_{1} (x) = x^{- 1} (1 + 10 x + 5 x^{2} + \frac{10 x^{3}}{9} + \dots), y_{2} (x) = x^{1 / 2} (1 + \frac{11 x}{20} - \frac{11 x^{2}}{224} + \frac{671 x^{3}}{24192} + \dots)$ $y_{1} (x) = x^{- 1} (1 + 10 x + 5 x^{2} + \frac{10 x^{3}}{9} + \dots), y_{2} (x) = x^{1 / 2} (1 + \frac{11 x}{20} - \frac{11 x^{2}}{224} + \frac{671 x^{3}}{24192} + \dots)$
34. $y_{1} (x) = x (1 - \frac{x^{2}}{42} + \frac{x^{4}}{1320} + \dots), y_{2} (x) = x^{- 1 / 2} (1 - \frac{7 x^{2}}{24} + \frac{19 x^{4}}{3200} + \dots)$ $y_{1} (x) = x (1 - \frac{x^{2}}{42} + \frac{x^{4}}{1320} + \dots), y_{2} (x) = x^{- 1 / 2} (1 - \frac{7 x^{2}}{24} + \frac{19 x^{4}}{3200} + \dots)$

..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.

Table of Contents for Section 11.3

Create new playlist

Sign In

Sign Up

Table of Contents for
Section 11.3