Section 11.4

5. $J_{4} (x) = \frac{1}{x^{2}} (x^{2} - 24) J_{0} (x) + \frac{8}{x^{3}} (6 - x^{2}) J_{1} (x)$ $J_{4} (x) = \frac{1}{x^{2}} (x^{2} - 24) J_{0} (x) + \frac{8}{x^{3}} (6 - x^{2}) J_{1} (x)$
10. 3
11. $x^{2} J_{1} (x) + x J_{0} (x) - \int J_{0} (x) d x + C$ $x^{2} J_{1} (x) + x J_{0} (x) - \int J_{0} (x) d x + C$
12. $(x^{3} - 4 x) J_{1} (x) + 2 x^{2} J_{0} (x) + C$ $(x^{3} - 4 x) J_{1} (x) + 2 x^{2} J_{0} (x) + C$
13. $(x^{4} - 9 x^{2}) J_{1} (x) + (3 x^{3} - 9 x) J_{0} (x) + 9 \int J_{0} (x) d x + C$ $(x^{4} - 9 x^{2}) J_{1} (x) + (3 x^{3} - 9 x) J_{0} (x) + 9 \int J_{0} (x) d x + C$
14. $- x J_{1} (x) + \int J_{0} (x) d x + C$ $- x J_{1} (x) + \int J_{0} (x) d x + C$
15. $2 x J_{1} (x) - x^{2} J_{0} (x) + C$ $2 x J_{1} (x) - x^{2} J_{0} (x) + C$
16. $3 x^{2} J_{1} (x) + (3 x - x^{3}) J_{0} (x) - 3 \int J_{0} (x) d x + C$ $3 x^{2} J_{1} (x) + (3 x - x^{3}) J_{0} (x) - 3 \int J_{0} (x) d x + C$
17. $(4 x^{3} - 16 x) J_{1} (x) + (8 x^{2} - x^{4}) J_{0} (x) + C$ $(4 x^{3} - 16 x) J_{1} (x) + (8 x^{2} - x^{4}) J_{0} (x) + C$
18. $- 2 J_{1} (x) + \int J_{0} (x) d x + C$ $- 2 J_{1} (x) + \int J_{0} (x) d x + C$
19. $y (x) = x [c_{1} J_{0} (x) + C_{2} Y_{0} (x)]$ $y (x) = x [c_{1} J_{0} (x) + C_{2} Y_{0} (x)]$
20. $y (x) = \frac{1}{x} [c_{1} J_{1} (x) + c_{2} Y_{1} (x)]$ $y (x) = \frac{1}{x} [c_{1} J_{1} (x) + c_{2} Y_{1} (x)]$
21. $y (x) = x [c_{1} J_{1 / 2} (3 x^{2}) + c_{2} J_{- 1 / 2} (3 x^{2})]$ $y (x) = x [c_{1} J_{1 / 2} (3 x^{2}) + c_{2} J_{- 1 / 2} (3 x^{2})]$
22. $y (x) = x^{3} [c_{1} J_{2} (2 x^{1 / 2}) + c_{2} Y_{2} (2 x^{1 / 2})]$ $y (x) = x^{3} [c_{1} J_{2} (2 x^{1 / 2}) + c_{2} Y_{2} (2 x^{1 / 2})]$
23. $y (x) = x^{- 1 / 3} [c_{1} J_{1 / 3} (\frac{1}{3} x^{3 / 2}) + c_{2} J_{- 1 / 3} (\frac{1}{3} x^{3 / 2})]$ $y (x) = x^{- 1 / 3} [c_{1} J_{1 / 3} (\frac{1}{3} x^{3 / 2}) + c_{2} J_{- 1 / 3} (\frac{1}{3} x^{3 / 2})]$
24. $y (x) = x^{- 1 / 4} [c_{1} J_{0} (2 x^{3 / 2}) + c_{2} Y_{0} (2 x^{3 / 2})]$ $y (x) = x^{- 1 / 4} [c_{1} J_{0} (2 x^{3 / 2}) + c_{2} Y_{0} (2 x^{3 / 2})]$
25. $y (x) = x^{- 1} [c_{1} J_{0} (x) + c_{2} Y_{0} (x)]$ $y (x) = x^{- 1} [c_{1} J_{0} (x) + c_{2} Y_{0} (x)]$
26. $y (x) = x^{2} [c_{1} J_{1} (4 x^{1 / 2}) + c_{2} Y_{1} (4 x^{1 / 2})]$ $y (x) = x^{2} [c_{1} J_{1} (4 x^{1 / 2}) + c_{2} Y_{1} (4 x^{1 / 2})]$
27. $y (x) = x^{1 / 2} [c_{1} J_{1 / 2} (2 x^{3 / 2}) + c_{2} J_{- 1 / 2} (2 x^{3 / 2})]$ $y (x) = x^{1 / 2} [c_{1} J_{1 / 2} (2 x^{3 / 2}) + c_{2} J_{- 1 / 2} (2 x^{3 / 2})]$
28. $y (x) = x^{- 1 / 4} [c_{1} J_{3 / 2} (\frac{2}{5} x^{5 / 2}) + c_{2} J_{- 3 / 2} (\frac{2}{5} x^{5 / 2})]$ $y (x) = x^{- 1 / 4} [c_{1} J_{3 / 2} (\frac{2}{5} x^{5 / 2}) + c_{2} J_{- 3 / 2} (\frac{2}{5} x^{5 / 2})]$
29. $y (x) = x^{1 / 2} [c_{1} J_{1 / 6} (\frac{1}{3} x^{3}) + c_{2} J_{- 1 / 6} (\frac{1}{3} x^{3})]$ $y (x) = x^{1 / 2} [c_{1} J_{1 / 6} (\frac{1}{3} x^{3}) + c_{2} J_{- 1 / 6} (\frac{1}{3} x^{3})]$
30. $y (x) = x^{1 / 2} [c_{1} J_{1 / 5} (\frac{4}{5} x^{5 / 2}) + c_{2} J_{- 1 / 5} (\frac{4}{5} x^{5 / 2})]$ $y (x) = x^{1 / 2} [c_{1} J_{1 / 5} (\frac{4}{5} x^{5 / 2}) + c_{2} J_{- 1 / 5} (\frac{4}{5} x^{5 / 2})]$

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Table of Contents for Section 11.4

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Table of Contents for
Section 11.4