第一類習題:級數解法
1. Find two power series solutions of the differential equation
x2 1
yxy y 0 about the ordinary point x0.【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 y3x2y6y x, y
0 0,
0 2y .【93 交大電子 8%】
3. Find the series solution of xyyx10 in power of
x1
.【94 輔仁電子 16%】4. y
2x2
y
x22x2
y0 , y
1 3 , y
1 1 , find series solution about x1 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.
y4xy
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
2 e 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%】
8. The Legendre equation is given as
1x2
y2xyn n
1
y0 when n is agiven 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 n0, 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
y2xy0 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
y2xy0, 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 x4, 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
yxy 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 x x x x
(B) 1 3 1 4 1 5
( ) 6 12 120
y x x x x x
(C)
0 n
n nx c
y , c1 0,
1
2 1 2
k k
k
c c
c k k
, k1, 2, 3,
(D) 1 3 1 4 1 5
( ) 3 15 60
y x x x x
(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 0ysiny ,
0y , 6 y
0 0 by power series method.【94 台大應 力 10%】14. 2yxyy 0, solve by power series method.【94 中興材料 10%】
15. yxy2x, 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. yyx3y0, solve by series method.【93 台大生物環境 15%】
19. 已知yx y2
x2
yx,y
0 2,y
0 1,試解之並至少寫出前 5 項。【93 台大生機 10%】
20. y2yx3y0, find the first nonzero terms of series solution about x0.【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%】
23. ytyy1 t 2, find series solution about t0.【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 x0 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.
xyyy0, 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 x4, 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y2 xy2yx。【90 北科土木 15%】
29. 若y
cosx y
0;y
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
yxy y x。【91 彰師光電 10%】31. 求下式之級數解:x2yyy 0。【91 交大土木 15%】
32. Consider an ordinary differential equation
0ya x yb x y
(1) Under what conditions will x0 be an ordinary point? Write a power series form for the solution y x
.(2) Under what conditions will x0 be a regular singular point? Write a possible power series form for the solution y x
.(3) Under what conditions will x0 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 x0 to solve
1x2
y2xy12y0.【88成大土木 15%】
34. Find the general solution about x0 expressed as yc1y1c2y2 for the differential equation y2xy0 . 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 yxyy0.【87 交大機械 15%】
36. Determine the first 5 nonzero terms of the power series solution about x0 for the initial value problem shown below:
yexy2y1;y
0 3, y
0 1.【87 台科電機 10%】37. Apply power series method to solve y3y2y0.【86 中山資訊 8%】
38. 求yxyy1x2在x0附近之解。【86 台科化工 20%】
39. Solve by power series of
1x2
y2xy0.【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 x x x
y
45360 ....
1 504
1 12
) 1
( 4 7 10
2 x x x x x
y
Can you verify the solutions are linearly independent?【91 雲科電機 10%】
41. Find a general solution of the Legendre’s equation:
1x2
y2xy2y0 on the interval 1 x1 using the power series method.【91 逢甲電機‧電子 20%】42. 請利用級數(即
0 1 ni 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: y4xy
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: y2xyy0.【90 清大工程科學 10%】
45. y2xy2y0, solve by series method.【90 北科高分子 10%】
46. Find general power series solution of yx2y 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y2xy2y0。【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
0n n y
d d
can be
transformed in Legendre’s equation by means of the substitution xcos.【86 成大土研所乙組】
51. Solve the following differential equation
1x2
y2xy12y0.【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 yis a function of x.【20%】
53. (1) What is Bessel’s equation of order n? Write down the solutions for n integer and ninteger.
(2) What is Legendre’s equation? Describe what you know about Legendre polynomials.【20%】
54.
1x2
yxyyx, find series solution.【91 彰師光電 10%】55. Please discuss the existence of y
x by series solution near the x0 according to the regularity of f
x , y f
x y0.(1) If f
x has ordinary point at x0(2) If f
x has regular singular point at x0 (3) If f
x has irregular singular point at x0(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: 4yx3y 【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y2xy。【90 交大土木 20%】
59. Solve
1x2
y2xy by power series for x0.【87 雲科電機 15%】
60. Using the power series method, solve y2y 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