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

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∞ ∞ ∞<br />

# ww w # <br />

5. x y 2xy 2y œ 0 Ê x n 2<br />

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

<br />

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 n<br />

2nc x n<br />

n n 2cnx 0<br />

nœ2 nœ1 nœ0<br />

Ê œ<br />

power of x coefficient equation<br />

0<br />

x 2c0 œ 0 Ê c0<br />

œ 0<br />

1<br />

x 21c 1 2c1<br />

œ 0 Ê 0œ<br />

0<br />

2<br />

x 21c 222c 2 2c2<br />

œ 0 Ê 0œ<br />

0<br />

3<br />

x 32c 323c 3 2c3 œ 0 Ê c3<br />

œ 0<br />

4<br />

x 43c 424c<br />

4 2c 4 œ 0 Ê c 4 œ 0<br />

ã ã ã<br />

n<br />

x nn 1c 2n c 2c œ 0 Ê c œ 0, n 3<br />

y œ c xc x<br />

1 2 2<br />

n n n n<br />

∞ ∞ ∞<br />

ww w <br />

6. y xy y œ 0 Ê n 2<br />

n n1 c x x n 1<br />

<br />

nc x n<br />

c x œ 0<br />

n n n<br />

nœ2 nœ1 nœ0<br />

∞ ∞ ∞<br />

n2 nn 1c x n<br />

nc x n<br />

n n cnx 0<br />

nœ2 nœ1 nœ0<br />

Ê œ<br />

power of x coefficient equation<br />

0<br />

1<br />

x 21c 2 c0 œ 0 Ê c2 œ 2c0<br />

1<br />

x 32c 3 1c 1 c1 œ 0 Ê c3<br />

œ 0<br />

2<br />

1 1<br />

x 43c 4 2c 2 c2 œ 0 Ê c4 œ<br />

12c2 œ 24c0<br />

3<br />

1<br />

x 54c 5 3c<br />

3 c3 œ 0 Ê c5 œ<br />

10c3<br />

œ 0<br />

4<br />

2 1<br />

x 65c 6 4c 4 c4 œ 0 Ê c6 œ<br />

15c4 œ 180c0<br />

ã ã ã<br />

n<br />

n1<br />

x n2n1c nc c œ 0 Ê c œ c<br />

n 2 n n n 2 n2n1<br />

n<br />

1 2 1 4 1 6<br />

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

1 x x x á‰<br />

cx<br />

0 1 2 0 24 0 180 0 0 2 24 180<br />

1<br />

ww<br />

7. 1 xy y 0 1 x<br />

n2 œ Ê nn 1c x n<br />

c x œ 0<br />

n<br />

n<br />

nœ2 nœ0<br />

∞ ∞ ∞<br />

n2 n1 n n<br />

n<br />

n<br />

nœ2 nœ2 nœ0<br />

Ê nn1c x nn1c x c x œ 0<br />

∞<br />

∞<br />

Section 17.5 Power-Series Soutions 1025<br />

power of x<br />

coefficient equation<br />

0<br />

1<br />

x 21c 2 c0 œ 0 Ê c2 œ<br />

2c0<br />

1<br />

1 1 1 1<br />

x 32c 3 21c 2 c1 œ 0 Ê c3 œ 3c2 <br />

6c1 œ<br />

6c1 <br />

6c0<br />

2<br />

1 1<br />

1 1<br />

x 43c 4 32c 3 c2 œ 0 Ê c4 œ 2c3<br />

12<br />

c2 œ 12<br />

c1 8<br />

c0<br />

3<br />

3 1 7 1<br />

x 54c 5 43c 4 c3 œ 0 Ê c5 œ 5c4 <br />

12c3 œ<br />

120c1 <br />

12c0<br />

4<br />

2 1 1 43<br />

x 65c 6 54c 5 c4 œ 0 Ê c6 œ 3c5 <br />

30c4 œ 24c1 <br />

720c0<br />

ã ã ã<br />

n<br />

x n2n1c n2 nn 1c n1 cn œ 0 Ê cn2 œ n cn 1 1<br />

n2 <br />

n2 n1<br />

cn<br />

1 2<br />

yœ c0 cx 1 cx 0 ˆ 1 1<br />

c1 c0‰ 3<br />

x ˆ 1 1<br />

c1 c0‰ 4<br />

x ˆ 7 1<br />

c1 c0‰ 5 1 43<br />

2 6 6 12 8 120 12<br />

x ˆ 6<br />

24c1 <br />

720c0‰<br />

x á<br />

œ c ˆ 1 2 1 3 1 4 1 5 43 6<br />

1 x x x x x በc ˆ 1 3 1 4 7 5 1 6<br />

x x x x x á‰<br />

0 2 6 8 12 720 1 6 12 120 24<br />

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