1. Ordinary point
2. Ordinary point
3. Irregular singular point
4. Irregular singular point
5. Regular singular point; r1=0, r2=−1
6. Regular singular point; r1=1, r2=−2
7. Regular singular point; r=−3, −3
8. Regular singular point; r=12, −3
9. Regular singular point x=1
10. Regular singular point x=1
11. Regular singular points x=1, −1
12. Irregular singular point x=2
13. Regular singular points x=2, −2
14. Irregular singular points x=3, −3
15. Regular singular point x=2
16. Irregular singular point x=0, regular singular point x=1
17. y1(x)=cos x−−√, y2(x)=sin x−−√
18. y1(x)=∑n=0∞xnn!(2n+1)!!, y2(x)=x−1/2∑n=0∞xnn!(2n−1)!!
19. y1(x)=x3/2(1+3∑n=1∞xnn!(2n+3)!!), y2(x)=1−x−∑n=2∞xnn!(2n−3)!!
20. y1(x)=x1/3∑n=0∞(−1)n2nxnn!⋅4⋅7⋯(3n+1), y2(x)=∑n=0∞(−1)n2nxnn!⋅2⋅5⋯(3n−1)
21. y1(x)=x(1+∑n=1∞x2nn!⋅7⋅11⋯(4n+3)), y2(x)=x−1/2(1+∑n=1∞x2nn!⋅1⋅5⋯(4n−3))
22. y1(x)=x3/2(1+∑n=1∞(−1)nx2nn!⋅9⋅13⋯(4n+5)), y2(x)=x−1(1+∑n=1∞(−1)n−1x2nn!⋅3⋅7⋯(4n−1))
23. y1(x)=x1/2(1+∑n=1∞x2n2n⋅n!⋅19⋅31⋯(12n+7)), y2(x)=x−2/3(1+∑n=1∞x2n2n⋅n!⋅5⋅17⋯(12n−7))
24. y1(x)=x1/3(1+∑n=1∞(−1)nx2n2n⋅n!⋅7⋅13⋯(6n+1)), y2(x)=1+∑n=1∞(−1)nx2n2n⋅n!⋅5⋅11⋯(6n+1)
25. y1(x)=x1/2∑n=0∞(−1)nxnn!⋅2n=x1/2e−x/2, y2(x)=1+∑n=1∞(−1)nxn(2n−1)!!
26. y1(x)=x1/2∑n=0∞x2nn!⋅2n=x1/2exp(12x2), y2(x)=1+∑n=1∞2nx2n3⋅7⋯(4n−1)
27. y1(x)=1xcos 3x, y2(x)=1xsin 3x
28. y1(x)=1x cosh 2x, y2(x)=1x sinh 2x
29. y1(x)=1xcos x2, y2(x)=1xsin x2
30. y1(x)=cos x2, y2(x)=sin x2
31. y1(x)=x1/2 cosh x, y2(x)=x1/2 sinh x
32. y1(x)=x+x25, y2(x)=x−1/2(1−5x2−15x28−5x348+⋯)
33. y1(x)=x−1(1+10x+5x2+10x39+⋯), y2(x)=x1/2(1+11x20−11x2224+671x324192+⋯)
34. y1(x)=x(1−x242+x41320+⋯), y2(x)=x−1/2(1−7x224+19x43200+⋯)