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

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1026 Chapter 17 Second-Order Differential Equations<br />

∞ ∞ ∞<br />

# ww w #<br />

8. 1 x y 4xy 6y 0 1 x n2 nn 1c x 4x n1 œ Ê n c x 6<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 />

4nc x n<br />

n n n 6cnx 0<br />

nœ2 nœ2 nœ1 nœ0<br />

Ê œ<br />

#<br />

power of x<br />

coefficient equation<br />

0<br />

x 21c2 6c0 œ 0 Ê c2 œ 3c0<br />

1<br />

1<br />

x 32c 3 41c 1 6c1 œ 0 Ê c3 œ 3c1<br />

2<br />

1<br />

x 43c 4 21c 2 42c 2 6c2 œ 0 Ê c4 œ<br />

3c2 œ c0<br />

3<br />

x 54c 5 32c 3 43c 3 6c3 œ 0 Ê c5 œ 3 c3 5<br />

œ 5c1<br />

4<br />

11 11<br />

x 65c 43c 44c 6c œ 0 Ê c œ<br />

15c œ 15c<br />

ã ã ã<br />

n<br />

x n2n1c nn 1c 4nc 6c œ 0 Ê c<br />

<br />

6 4 4 4 6 4 0<br />

n2 n n n n2<br />

œ n 3n 6<br />

n2n1 c n<br />

2 1 3 4 1 5 11 6<br />

y œ c0 c1x3c0x <br />

3c1x c0x <br />

5c1x <br />

15c0x<br />

á<br />

œ c ˆ 2 4<br />

13x x 11 6<br />

x በc ˆ 3 5<br />

x 1 x 1 x á‰<br />

0 15 1 3 5<br />

∞ ∞ ∞<br />

# ww w #<br />

9. x 1y 2xy 2y 0 x 1 n2 nn 1c x 2x n1 œ Ê n c x 2<br />

n<br />

c x œ 0<br />

n n n<br />

nœ2 nœ1 nœ0<br />

∞ ∞ ∞ ∞<br />

n<br />

nn 1c x n2 nn 1c x n<br />

2nc x n<br />

n n n 2cnx 0<br />

nœ2 nœ2 nœ1 nœ0<br />

Ê œ<br />

power of x<br />

coefficient equation<br />

0<br />

x 21c 2 2c0 œ 0 Ê c2 œ c0<br />

1<br />

x 32c 3 21c 1 2c1 œ 0 Ê c3<br />

œ !<br />

2<br />

1 1<br />

x 21c 43c 22c 2c œ 0 Ê c œ<br />

3c œ 3c<br />

3<br />

1<br />

x 32c 3 54c 5 23c 3 2c3 œ 0 Ê c5 œ<br />

2c3<br />

œ !<br />

4<br />

3 1<br />

x 43c 65c 24c 2c œ 0 Ê c œ<br />

5c œ 5c<br />

ã ã ã<br />

n<br />

x nn 1cn n 2n 1cn2 2ncn 2cn œ 0 Ê cn2<br />

œ<br />

n1<br />

2 4 2 2 4 2 0<br />

4 6 4 4 6 4 0<br />

n1 c n<br />

2 1 4 1 6<br />

yœ c cxcx cx cx á œ cˆ 2 1 4 1 6<br />

1x x x á‰<br />

cx<br />

0 1 0 3 0 5 0 0 3 5<br />

1<br />

∞ ∞ ∞<br />

ww w # #<br />

10. y y x y œ 0 Ê n 2<br />

n n 1 c x n 1<br />

<br />

n c x x n<br />

c x œ 0<br />

n n n<br />

nœ2 nœ1 nœ0<br />

∞ ∞ ∞<br />

n2 nn 1c x n1 n ncnx n2<br />

cnx 0<br />

nœ2 nœ1 nœ0<br />

Ê œ<br />

power of x<br />

coefficient equation<br />

0<br />

1<br />

x 21c 2 1c 1 œ 0 Ê c2 œ 2c1<br />

1<br />

1 1<br />

x 32c 3 2c 2 œ 0 Ê c3 œ 3c2 œ<br />

6c1<br />

2<br />

1 1 1 1<br />

x 43c 4 3c 3 c0 œ 0 Ê c4 œ 4c3 <br />

12c0 œ 24c1<br />

<br />

12c<br />

0<br />

3<br />

1 1 1 1<br />

x 54c 5 4c 4 c1 œ 0 Ê c5 œ 5c4 <br />

20c1 œ<br />

120c1 <br />

30c0<br />

4<br />

1 1 13 1<br />

x 65c 5c c œ 0 Ê c œ 6c <br />

30c œ 720c <br />

180c<br />

ã ã ã<br />

n<br />

x n2n1cn2<br />

n1cn1 cn2 œ 0 Ê cn2 œ 1 cn1 <br />

1 cn2<br />

6 5 2 6 5 2 1 0<br />

n2 n2n1<br />

1 2 1 3<br />

yœ c0 cx 1 cx 0 cx 1 ˆ 1 1<br />

c1 c0‰ 4<br />

x ˆ 1 1<br />

c1 c0‰ 5 13 1<br />

2 6 24 12 120 30<br />

x ˆ 6<br />

720c1 <br />

180c0‰<br />

x á<br />

œ c ˆ 1 2 1 4 1 5 1 6<br />

1 x x x x በc ˆ 1 3 1 4 1 5 13 6<br />

x x x x x á‰<br />

0 2 12 30 180 1 6 24 120 720<br />

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