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8.4.1: Series Solutions About a Regular Singular Point (Exercises)

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    In Exercises 1-60 find the series in x for the general solution.

    1. \(2xy''-y'+2y=0\)

    2. \(4xy''+{1\over 2}y'+y=0\)

    3. \(3xy''+(2-x)y'-y=0\)

    4. \(2xy''-(3+2x)y'+y=0\)

    5. \(9x^2y''+9x^2y'+2y=0\)

    6. \(xy''+2y'-xy=0\)

    7. \(xy''-xy'+y=0\)

    8. \(xy''+(1-x)y'-y=0\)

    9. \(2xy''+5y'+xy=0\)

    10. \(2x^2y''-xy'+(x^2+1)y=0\)

    11. \(x^2y''-(x-{2\over 9})y=0\)

    12. \(x^2y''+xy'+(x^2-{4\over 9})y=0\)

    13. \(2x^2y''+3xy'+(2x-1)y=0\)

    14. \(x^2y''+xy'+(x^2-{1\over 4})y=0\)

    15. \(y''+{3\over x}y'-2y=0\)

    16. \(xy''+y'+y=0\)

    17. \(2x^2(1+x+x^2)y''+x(3+3x+5x^2)y'-y=0\)

    18. \(3x^2y''+2x(1+x-2x^2)y'+(2x-8x^2)y=0\)

    19. \(x^2(3+3x+x^2)y''+x(5+8x+7x^2)y'-(1-2x-9x^2)y=0\)

    20. \(4x^2y''+x(7+2x+4x^2)y'-(1-4x-7x^2)y=0\)

    21. \(12x^2(1+x)y''+x(11+35x+3x^2)y'-(1-10x-5x^2)y=0\)

    22. \(x^2(5+x+10x^2)y''+x(4+3x+48x^2)y'+(x+36x^2)y=0\)

    23. \(8x^2y''-2x(3-4x-x^2)y'+(3+6x+x^2)y=0\)

    24. \(18x^2(1+x)y''+3x(5+11x+x^2)y'-(1-2x-5x^2)y=0\)

    25. \(x(3+x+x^2)y''+(4+x-x^2)y'+xy=0\)

    26. \(10x^2(1+x+2x^2)y''+x(13+13x+66x^2)y'-(1+4x+10x^2)y=0\)

    27. \(x^2(8+x)y''+x(2+3x)y'+(1+x)y=0\)

    28. \(x^2(3+4x)y''+x(11+4x)y'-(3+4x)y=0\)

    29. \(2x^2(2+3x)y''+x(4+11x)y'-(1-x)y=0\)

    30. \(x^2(2+x)y''+5x(1-x)y'-(2-8x)y\)

    31. \(x^2(6+x)y''+x(11+4x)y'+(1+2x)y=0\)

    32. \(6x^2y''+x(1+6x^2)y'+(1+9x^2)y=0\)

    33. \(x^2(8+x^2)y''+7x(2+x^2)y'-(2-9x^2)y=0\)

    34. \(9x^2(1+x^2)y''+3x(3+13x^2)y'-(1-25x^2)y=0\)

    35. \(4x^2(1+x^2)y''+4x(1+6x^2)y'-(1-25x^2)y=0\)

    36. \(8x^2(1+2x^2)y''+2x(5+34x^2)y'-(1-30x^2)y=0\)

    37. \(x^2y''-x(1-x)y'+(1-x^2)y=0\)

    38. \(x^2(1+x+2x^2)y'+x(3+6x+7x^2)y'+(1+6x-3x^2)y=0\)

    39. \(x^2(1+2x+x^2)y''+x(1+3x+4x^2)y'-x(1-2x)y=0\)

    40. \(4x^2(1+x+x^2)y''+12x^2(1+x)y'+(1+3x+3x^2)y=0\)

    41. \(x^2(1+x+x^2)y''-x(1-4x-2x^2)y'+y=0\)

    42. \(9x^2y''+3x(5+3x-2x^2)y'+(1+12x-14x^2)y=0\)

    43. \(x^2y''+x(1+x+x^2)y'+x(2-x)y=0\)

    44. \(x^2(1+2x)y''+x(5+14x+3x^2)y'+(4+18x+12x^2)y=0\)

    45. \(4x^2y''+2x(4+x+x^2)y'+(1+5x+3x^2)y=0\)

    46. \(16x^2y''+4x(6+x+2x^2)y'+(1+5x+18x^2)y=0\)

    47. \(9x^2(1+x)y''+3x(5+11x-x^2)y'+(1+16x-7x^2)y=0\)

    48. \(x^2(1+2x)y''+x(5+9x)y'+(4+3x)y=0\)

    49. \(x^2(1-2x)y''-x(5+4x)y'+(9+4x)y=0\)

    50. \(x^2(1+4x)y''-x(1-4x)y'+(1+x)y=0\)

    51. \(x^2(1+x)y''+x(1+2x)y'+xy=0\)

    52. \(x^2(1-x)y''+x(7+x)y'+(9-x)y=0\)

    53. \(x^2(1+x^2)y''+x(3+8x^2)y'+(1+12x^2)y\)

    54. \(x^2y''-x(1-x^2)y'+(1+x^2)y=0\)

    55. \(x^2(1-2x^2)y''+x(5-9x^2)y'+(4-3x^2)y=0\)

    56. \(x^2(2+x^2)y''+x(14-x^2)y'+2(9+x^2)y=0\)

    57. \(x^2(1+x^2)y''+x(3+7x^2)y'+(1+8x^2)y=0\)

    58. \(x^2(1-2x)y''+3xy'+(1+4x)y=0\)

    59. \(x(1+x)y''+(1-x)y'+y=0\)

    60. \(x^2(1+x^2)y''-x(1+9x^2)y'+(1+25x^2)y=0\)

    This page titled 8.4.1: Series Solutions About a Regular Singular Point (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench via source content that was edited to the style and standards of the LibreTexts platform.