Thomas Calculus 13th [Solutions]

1024 Chapter 17 Second-Order Differential Equations
∞ ∞ ∞ ∞
ww
3. y 4y œ 0 Ê n 2
n n 1 c x 4 n
c x œ 0 Ê n 2
nn 1c x n
4c x œ 0
n n n n
nœ2 nœ0 nœ2 nœ0
power of x coefficient equation
0
x 21c2 4c0 œ 0 Ê c2 œ 2c0
1
2
x 32c 3 4c1 œ 0 Ê c3 œ 3c1
2
1 2
x 43c 4 4c2 œ 0 Ê c4 œ 3c2 œ
3c0
3
1 2
x 54c 5 4c3 œ 0 Ê c5 œ 5c3
œ
15c
1
4
2 4
x 65c 4c œ 0 Ê c œ 15c œ 45c
ã ã ã
n
4
x n 2n 1c 4c œ 0 Ê c œ n2n1c
6 4 6 4 0
n 2 n n 2 n
0 1 0
2 2 3 2 4 2 5 4
3 1 3 0 15 1 45 0
6
0 0
2 2 4 4 6 2 3 2
3 0 45 0 1 3 1 15 1
5
0
2 4 6 3 5 ∞ n ∞ n
2x 2x 2x 2x 2x 1 2n 1 2n1
2x 4x 6x 2 3x 5x 0 2nx 2 2n1x
nœ0 nœ0
0
c1 c1
2 0
2
y œ c c x2c x c x c x c x c x á
œ c 2cx cx cx á cx cx cx á
c1 c
œ c Š 1 á‹ Š 2x á‹ œ c
1
2x
2x
œ c cos 2x sin 2x œ a cos 2x b sin 2x, where a œ c and b œ
∞ ∞ ∞
ww w
4. y 3 y 2y œ 0 Ê n 2
n n 1 c x 3 n 1
n c x 2
n
c x œ 0
n n n
nœ2 nœ1 nœ0
∞ ∞ ∞
n2 nn 1c x n1 n 3ncnx n
2cnx 0
nœ2 nœ1 nœ0
Ê œ
power of x
coefficient equation
0
3
x 21c 2 31c 1 2c0 œ 0 Ê c2 œ
2c1 c0
1
1 7
x 32c 3 32c 2 2c1 œ 0 Ê c3 œ c2
3c1 œ
6c1 c0
2
3 1 5 7
x 43c 4 33c 3 2c2 œ 0 Ê c4
œ
4c3
6c2 œ
8c1
12c0
3
3 1 31 1
x 54c 5 34c 4 2c3 œ 0 Ê c5 œ
5c4
10c3 œ
120c1
4c0
4
1 1 7 31
x 65c 35c 2c œ 0 Ê c œ
2c
15c œ
80c
360c
ã ã ã
n
x n
2n1c 3n1c 2c œ 0 Ê c œ 3 c
2 c
6 5 4 6 5 4 1 0
n2 n1 n n2 n2 n1 n2n1
n
y œ c c xˆ 3c c ‰ 2
x ˆ 7c c ‰ 3
x ˆ 5 7
c c ‰ 4
x ˆ 31 1 5 7 31 6
c c ‰ x ˆ c c ‰ x á
0 1 2 1 0 6 1 0 8 1 12 0 120 1 4 0 80 1 360 0
2 3 7 4 1 5 31 6 3 2 7 3 5 4 31 5 7 6
0 0 0 12 0 4 0 360 0 1 2 1 6 1 8 1 120 1 80 1
0ˆ 2 1 3 7 4 1 5 31 6 ‰ 2 3 4 5 6
1ˆ 3 7 5 31 7 ‰
3 12 4 360 2 6 8 120 80
x 2x
œ œ ∞
x
∞
n
2x n
nx
nx
nœ0 nœ0
3 2 7 3 5 7 4 31 1 5
0 1 2 1 0 6 1 0 8 1 12 0 120 1 4 0
œ c c x c x c x c x c x á c x c x c x c x c x c x á
œ c 1x x x x x á c x x x x x x á
Note that if we use the techniques of Section 17.2, our solution is y a e b e a b . Thus
aba2bx á œ c c xˆ c c ‰ x ˆ c c ‰ x ˆ c c ‰ x ˆ c c ‰ x á
If the series are equivalent, then take c0 œ a b and c1 œ a 2b and substitute in our first series for c 0 and c 1 to obtain
y œ a ba 2bx ˆ 3a 2ba b‰ 2
x ˆ 7a 2ba b‰ 3
x ˆ 5 7
a 2b a b‰
4
2 6 8 12
x
ˆ 31 1
a2b ab ‰ 5
x ˆ 7 31
a2b ab‰
6
120 4 80 360
x á
œ aba2bxˆ 1a2b‰ 2
x ˆ 1 4
a b‰ 3
x ˆ 1 2
a b‰ 4
x ˆ 1 4
a b‰ 5
x ˆ 1 4
a b‰
6
x á
2 6 3 24 3 120 15 720 45
1 2 1 3 1 4 1 5 1 6 2 4 3 2 4 4 5 4 6
2 6 24 120 720 3 3 15 45
1 2 1 3 1 4 1 5 1 6
2
2x 3
2x 4
2x 5
2x 6
2x
2x 3x 4x 5x 6x 2x 3x 4x 5x 6x
œ a ax ax ax ax ax ax á b 2bx 2bx bx bx bx bx á
œ aˆ 1x x x x x x በbŠ 12x á‹
∞ ∞ n
2x x 2x
nx
nœ0 nœ0
nx
n
œ a x
b
œ ae be
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