Thomas Calculus 13th [Solutions]
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Thomas Calculus 13th [Solutions]

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1028 Chapter 17 Second-Order Differential Equations<br />

∞ ∞ ∞<br />

ww w <br />

14. y 2xy 4y œ 0 Ê n 2<br />

n n 1 c x 2x n 1<br />

<br />

n c x 4<br />

n<br />

c x œ 0<br />

n n n<br />

nœ2 nœ1 nœ0<br />

∞ ∞ ∞<br />

n2 nn 1c x n<br />

2nc x n<br />

n n 4cnx 0<br />

nœ2 nœ1 nœ0<br />

Ê œ<br />

power of x coefficient equation<br />

0<br />

x 21c2 4c0 œ 0 Ê c2 œ 2c0<br />

1<br />

1<br />

x 32c 3 21c 1 4c1 œ 0 Ê c3 œ 3c1<br />

2<br />

x 43c 4 22c 2 4c2 œ 0 Ê c4<br />

œ 0<br />

3<br />

x 54c 5 23c 3 4c<br />

3 œ 0 Ê c 1 1<br />

5 œ<br />

10c 3 œ 30c<br />

1<br />

4<br />

2<br />

x 65c 6 24c 4 4c4 œ 0 Ê c6 œ<br />

15c4<br />

œ 0<br />

ã ã ã<br />

n<br />

2n 1<br />

n 2 n n n 2 n2n1<br />

n<br />

x n2n1c 2nc 4c œ 0 Ê c œ c<br />

2 1 3 1 5 2 1 3 1 5<br />

y œ c c x 2c x c x c x á œ c 1 2x c ˆ x x x á‰<br />

0 1 0 3 1 30 1 0 1 3 30<br />

∞ ∞ ∞<br />

ww w <br />

15. y 2xy 3y œ 0 Ê n 2<br />

n n 1 c x 2x n 1<br />

<br />

n c x 3<br />

n<br />

c x œ 0<br />

n n n<br />

nœ2 nœ1 nœ0<br />

∞ ∞ ∞<br />

n2 nn 1c x n<br />

2nc x n<br />

n n 3cnx 0<br />

nœ2 nœ1 nœ0<br />

Ê œ<br />

power of x coefficient equation<br />

0<br />

3<br />

x 21c 2 3c0 œ 0 Ê c2 œ 2c0<br />

1<br />

1<br />

x 32c 3 21c 1 3c1 œ 0 Ê c3 œ 6c1<br />

2<br />

1 1<br />

x 43c 4 22c 2 3c2 œ 0 Ê c4 œ<br />

12c2 œ 8c0<br />

3<br />

x<br />

54c 23c 3c œ 0 Ê c œ 3 1<br />

20c œ 40c<br />

4<br />

1 1<br />

x 65c 6 24c 4 3c4 œ 0 Ê c6 œ<br />

6c4 œ 48c0<br />

ã ã ã<br />

n<br />

2n3<br />

x n2n1c 2nc 3c œ 0 Ê c œ c<br />

5 3 3 5 3 1<br />

n 2 n n n 2 n2n1<br />

n<br />

3 2 1 3 1 4 1 5 1 6<br />

yœ c0 cx 1 <br />

2cx 0 <br />

6cx 1 <br />

8cx 0 <br />

40cx 1 <br />

48cx<br />

0 á<br />

œ c ˆ 3 2 1 4 1 6<br />

1 x x x በc ˆ 1 3 1 5<br />

x x x á‰<br />

0 2 8 48 1 6 40<br />

∞ ∞ ∞<br />

# ww w #<br />

16. 1 x y xy 4y 0 1 x n2 nn 1c x x n1 œ Ê n c x 4<br />

n<br />

c x œ 0<br />

n n n<br />

nœ2 nœ1 nœ0<br />

∞ ∞ ∞ ∞<br />

n2 nn 1c x n<br />

nn 1c x n<br />

nc x n<br />

n n n 4cnx 0<br />

nœ2 nœ2 nœ1 nœ0<br />

Ê œ<br />

power of x<br />

coefficient equation<br />

0<br />

x 21c2 4c0 œ 0 Ê c2 œ 2c0<br />

1<br />

1<br />

x 32c 3 1c 1 4c1 œ 0 Ê c3 œ 2c1<br />

2<br />

x 43c 4 21c 2 2c 2 4c2 œ 0 Ê c4<br />

œ 0<br />

3<br />

x 54c 5 32 c 3c 4c œ 0 Ê c œ 1 c 4<br />

œ 8c<br />

4<br />

2<br />

x 65c 6 43c 4 4c 4 4c4 œ 0 Ê c6 œ<br />

5c4<br />

œ 0<br />

ã ã ã<br />

n<br />

x n2n1c nn 1c nc 4c œ 0 Ê c œ<br />

n2<br />

c<br />

3 3 3 5 3 1<br />

n2 n n n n2 n1<br />

n<br />

2 1 1 5 2 1 3 1 5<br />

y œ c c x2c x c x c x á œ c 1x c ˆ x x x á‰<br />

0 1 0 2 1 8 1 0 1 2 8<br />

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