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(1)

第一類習題:級數解法

1. Find two power series solutions of the differential equation

x2 1

yxy y 0 about the ordinary point x0.【94 中興機械 20%,93 成大電機 20%】

2. Using power series method,

0 n n n

y c x

, to solve y3x2y6yx, y

 

0 0,

 

0 2

y .【93 交大電子 8%】

3. Find the series solution of xyyx10 in power of

x1

.【94 輔仁電子 16%】

4. y

2x2

y

x22x2

y0 , y

 

1 3 , y

 

1 1 , find series solution about x1 at least four nonzero terms.【92 元智電機 30%】

5. Find a power series solution in powers of x of the following differential equation.

y4xy

4x2 2

y0【94 淡江化工 25%】

6. 單選題,每題恰有一解,答對一小題給 3 分,答錯或不答,不給分也不扣分。

For the IVP

 

   

 

 

 

 

2 1

2

1

2 3 3 2 0

2 20, 2 2

x x y y x y

y y

      



   

 , the power-series solution

about the initial point is

   

0

2 n

n n

y x a x

. Then

(1) a0  (A)1 (B)-1 (C)2 (D)-2 (E)none (3%) (2) a1 (A)1 (B)-1 (C)2 (D)-2 (E)none (3%) (3) a2  (A)1 (B)-1 (C)2 (D)-2 (E)none (3%)

(4) a3  (A)1 (B)-1 (C)2 (D)-2 (E)none (3%) 【94 交大電機】

7. For the following equation,

2 0

2

2e y

dx y

d x

Please find the solution based on the power series method and write out first 5 non-zero terms in the solution.【93 清大電機 10%】

(2)

8. The Legendre equation is given as

1x2

y2xyn n

1

y0 when n is a

given number. Use power series method

0 m

m mx a

y to solve the ODE.

(1) Derive the recurrence equation. (5%) (2) Express a2,a4 in terms of a . (3%) 0 (3) Express a ,3 a in terms of 5 a1. (3%)

(4) Find the general solution ya0y1a1y2. (3%) (5) Let n0, find y1 and y2. (3%)

(6) Prove that y2 in (5) can be written as

x y x

  1 ln1 2 1

2 . (3%)

(7) Solve y

1x2

y2xy0 by z y, compare the solution in (6). (10%) 【93 清大電機 10%】

9. Use the Maclaurin series to solve the general solution.

x21

y2xy0, y

 

0 0, y

 

0 1.【93 台大電機 7%】

10. Given

x1

y y 2

x1

y0,

(1) Find two linearly independent power series solutions with center x4, with at least four nonzero terms for each series solution. Justify the solutions are linearly independent.

(2) Solve for y

 

4 1, y

 

4 1.【90 元智電機控制組 30%】

11.

1 x 2

yxy y x, find series solution.【91 彰師光電 10%】

12. 以下的答案中,那些是y  

1 x y

0的解。(複選)

(A) 1 2 1 3 1 4 1 5

( ) 1

2 6 18 36

y x   xxxx 

(B) 1 3 1 4 1 5

( ) 6 12 120

y x  x xxx 

(C)

0 n

n nx c

y , c1 0,

 

1

2 1 2

k k

k

c c

c k k

 

  , k1, 2, 3,

(3)

(D) 1 3 1 4 1 5

( ) 3 15 60

y xxxx 

(E) y

cnxn, 0c0 , ck2

kck1



ckk1 2

, 1, 2, 3,k

(F) 以上皆非【91 台大電機 5%】

13. Solve 0ysiny ,

 

0

y  , 6 y

 

0 0 by power series method.【94 台大應 力 10%】

14. 2yxyy 0, solve by power series method.【94 中興材料 10%】

15. yxy2x, y

 

0 3, y

 

1 0, solve by series method. 【92 淡江化工 15%】

16. Solve y  

1 x x22x3

y3y3x5x2.【94 交大土木 20%】

17. y y 0, solve by series method.【92 海洋光電 16%】

18. yyx3y0, solve by series method.【93 台大生物環境 15%】

19. 已知yx y2

x2

yxy

 

0 2y

 

0 1,試解之並至少寫出前 5 項。

【93 台大生機 10%】

20. y2yx3y0, find the first nonzero terms of series solution about x0.【93 台科電機 10%】

21. y sinx y  solve by series method.【93 中正電機 4%】 0 22. By using series expansion, find a solution for equation:

(1) xy

dx y

d22  with x. (7%)

(2) 2 2 0

2   y

dy xdy dx

y

d  ,  x, and  being a constant. (8%)【93 海洋 光電 15%】

(4)

23. ytyy1 t2, find series solution about t0.【93 元智光電 20%】

24. Solve y xy 4, y

 

1 2, y

 

1 0.【92 嘉義生機 10%】

25.

x2 1

y  y y 0, find series solution about x0 at least up to x .【92 交4 大電信 10%】

26. Find the power series solution of the following initial value problem about x = 1.

xyyy0, y

 

1 2.【88 雲科機械 15%】

27. Given

x1

y y 2

x1

y0.

(1) Find two linearly independent power series solutions with center x4, with at least four nonzero terms for each series solution. Justify the solutions are linearly independent.

(2) Solve for y

 

4 1, y

 

4 1.【90 元智電機控制組 30%】

28. 試以冪級數求解y2 xy2yx。【90 北科土木 15%】

29. 若y 

cosx y

0y

 

2 2, y

 

2 1, 試求其冪級數(Power series)之前 四項。【89 北科土木 15%】

30. 以級數展開法,解

1 x 2

yxy y x【91 彰師光電 10%】

31. 求下式之級數解:x2yyy 0。【91 交大土木 15%】

32. Consider an ordinary differential equation

   

0

ya x yb x y

(1) Under what conditions will x0 be an ordinary point? Write a power series form for the solution y x

 

.

(5)

(2) Under what conditions will x0 be a regular singular point? Write a possible power series form for the solution y x

 

.

(3) Under what conditions will x0 be an irregular singular point? Write a possible power series form for the solution y x

 

.【86 清大動機 15%】

33. Using power series method about x0 to solve

1x2

y2xy12y0.【88

成大土木 15%】

34. Find the general solution about x0 expressed as yc1y1c2y2 for the differential equation y2xy0 . Show that y1 and y2 are linearly independent. Find the interval of convergence for this solution.【87 交大電子 6%】

35. Use power series method to solve yxyy0.【87 交大機械 15%】

36. Determine the first 5 nonzero terms of the power series solution about x0 for the initial value problem shown below:

yexy2y1;y

 

0  3, y

 

0 1.【87 台科電機 10%】

37. Apply power series method to solve y3y2y0.【86 中山資訊 8%】

38. 求yxyy1x2x0附近之解。【86 台科化工 20%】

39. Solve by power series of

1x2

y2xy0.【86 交大機械 10%】

40. There are two solutions that are solved for the equation, y xy 0, in the power series

12960 ....

1 180

1 6

1 1 )

( 3 6 9

1 x   xxx

y

45360 ....

1 504

1 12

) 1

( 4 7 10

2 x xxxx

y

Can you verify the solutions are linearly independent?【91 雲科電機 10%】

(6)

41. Find a general solution of the Legendre’s equation:

1x2

y2xy2y0 on the interval 1 x1 using the power series method.【91 逢甲電機‧電子 20%】

42. 請利用級數(即

 

0 1 n

i i i

y x a a x

 

)展開方式解y y 0【91 高科環安 10%】

43. Solve the following second–order differential equation for y as a power series in powers of

xx0

where x0 0: y4xy

4x22

y0.【91 清大工程科

學 15%】

44. Use power series method to solve the following problem, find at least five terms of a general solution: y2xyy0.【90 清大工程科學 10%】

45. y2xy2y0, solve by series method.【90 北科高分子 10%】

46. Find general power series solution of yx2y 0.【89 交大環工 15%】

47.

x1

y y 2

x1

y0, y

 

4 5, y

 

4 0, 4 x。求級數解,只

需求最前面 5 項即可。【91 高科機械 20%】

48. 試以冪級數求解y2xy2y0。【90 北科大土研所】

49. 試以級數解求解xy y 0,並求該解之收斂半徑。【92 交大土研所甲組】

50. Show that the equation sin d y22 cos dy

1 sin

 

0

n n y

d d

  

can be

transformed in Legendre’s equation by means of the substitution xcos.【86 成大土研所乙組】

51. Solve the following differential equation

1x2

y2xy12y0.【88 成大土

研所丁組】

52. 試求解二階微分方程式y  

1 x x22x3

y3y3x5x2的通解,where y

(7)

is a function of x.【20%】

53. (1) What is Bessel’s equation of order n? Write down the solutions for n integer and ninteger.

(2) What is Legendre’s equation? Describe what you know about Legendre polynomials.【20%】

54.

1x2

yxyyx, find series solution.【91 彰師光電 10%】

55. Please discuss the existence of y

 

x by series solution near the x0 according to the regularity of f

 

x , y f

 

x y0.

(1) If f

 

x has ordinary point at x0

(2) If f

 

x has regular singular point at x0 (3) If f

 

x has irregular singular point at x0

(4) If the above ordinary equation change to y f

 

x y0. What is the different result with (2)?【87 北科電機 20%】

56. Use the Maclaurin series to solve the general solution including the recurrence relation: 4yx3y  【88 雲科電機 15%】

57. (1) Using power series to solve y ky 0, in which k is constant.

(2) Is the series from (1) equal to ekx? Why?【90 交大電機 18%】

58. 試以 Power Series Method 解y2xy。【90 交大土木 20%】

59. Solve

1x2

y2xy by power series for x0.【87 雲科電機 15%】

60. Using the power series method, solve y2y as a power series in powers of

1

x .【86 中正電機 10%】

61. The interval of convergence of the power series

 

1

3 2

n

n n

x n

is (1) (0,6) (2) [0,6] (3) [1,5] (4) (1,5) 【87 台大電機 5%】

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