Page 831 - Advanced_Engineering_Mathematics o'neil

Answers to Selected Problems 811

1
5. a 0 and a 1 arbitrary; a 2 = (3 − a 0 ),
2
n − 1
a n+2 = a n for n = 1,2,··· ,
(n + 1)(n + 2)
and

y(x) =a 0 + a 1 x
1 1 3 3(5) 3(5)(7)

4
2
6
10
8
+ (3 − a 0 ) x + x + x + x + x +···
2! 4! 6! 8! 10!
7. a 0 , a 1 are arbitrary; a 2 + a 0 = 0, 6a 3 + 2a 1 = 1,
(n − 3)a n−3 − 2a n−2
a n = for n = 4,5,··· ,
n(n − 1)
and

1 1 1
2 4 5 6
y(x) =a 0 1 − x + x − x − x + ···
6 10 90
1 1 1 7

5
3
4
6
+ a 1 x − x + x + x − x + ···
3 12 30 180
1 1 1 1 1

3
6
8
5
7
x −− x + x + x − x + ···
6 60 60 1260 480
with a 0 = y(0) and a 1 = y (0). The third bracket is a particular solution for the case a 0 = a 1 = 0.

9. a 0 , a 1 arbitrary; 2a 2 + a 1 + 2a 0 = 1, 6a 3 + 2a 2 + a 1 = 0, 12a 4 + 3a 3 =−1,
−(n − 1)a n−1 + (n − 4)a n−2
a n = for n = 5,6,···
n(n − 1)
and
1 1 1

2 3 4 5
y(x) =a 0 1 − x + x − x + x −···
3 12 30

1 1 1
2
+ a 1 x − x 2 + x − x 3
2 2 6
1 1 1
6
7
7
− x − x + x +···
24 360 2520
Section 4.2 Frobenius Solutions
1. y 1 (x) = c 0 (1 − x),

1
y 2 (x) =c ∗ (1 − x)ln(x) + 3x + x 2
0
4
1 1 1
5
4
3
+ x + x + x +···
36 288 2400
4 5 6 7
3. y 1 (x)= c 0 [x + 2x + 3x + 4x +···]
x 4 3 − 4x
= c 0 , y 2 (x) = c 0 ∗
(1 − x) 2 (1 − x) 2
1 1

5. y 1 (x) = c 0 x 1/2 − x 3/2 + x 5/2
2(1!)(3) 2 (2!)(3)(5)
2
1 1
− x 7/2 + x 9/2 +···
4
3
2 (3!)(3)(5)(7) 2 (4!)(3)(5)(7)(9)

n
(−1)
= c 0 x 1/2 1 + ∞ x n ,

n=1
2 n!(3 · 5···(2n + 1))
n
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October 14, 2010 17:50 THM/NEIL Page-811 27410_25_Ans_p801-866