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Chapter 3 51

# Chapter 3 51 - SECTION3J HIGHER DERIVATIVES I 201 29 9x2...

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Unformatted text preview: SECTION3J HIGHER DERIVATIVES I: 201 29. 9x2 +3,2 2 9 2 18m+2yy’ = 0 => 2yy’ = —18x => y/ : —9a:/y => 2 2 //_ 9 y'l—m‘y, ﬁ_9 WEE!) :_g.u9_\$:_9.% [sincexandymustsatisfy y _ _ 312 _ 92 313 y the original equation, 9:32 + y2 : 9]. Thus, y" = ——81/y3. yI 30-ﬁ+¢z7=1 :5 517+2ﬁ=0 :5 y/=_\/W; : ~ ﬁll/(WW-M/WH MWZHf/ﬁ y “—‘—m“— 2x 95 : M = 1 since as and y must satisfy the original equation, V; + ﬂ : 1. 25c \/:E 2:0 ﬂ 3132 31.\$3+y3=1 => 3\$2+3y2yI=0 => y’Zﬁgﬁ :> ,, _ y2(2\$) — \$2 - Zyy’ _ _2:cy2 — 2x2y(—\$2/y2) _ 2mg4 + 2\$4y : 2xy(y3 +x3) h _2_1‘1 y _ _ (.742)2 _ y4 216 y“ 2/5 since cc and y must satisfy the original equation, \$3 + y3 : 1. 32. m4 +y4 = a4 => 4x3 +4y3y' = 0 => 4y3 ’ : —4a:3 => y' = ems/313 => I 4 4 4 _ 4 2 y” I _ 243 30:2 — :63 ~ 392.14 : _3x2y2 . y e gc(~:1c‘°7y3) : _3x2 . 14 +30: 2 _3m2 . a_7 : 307\$ (313V 116 2142; y y 33. my) : .73" ;> f/(m) 2 mm ;» f”(\$) = ”(n _ 1)x"_2 => : fI")(;1,-) : n(n — 1)(n v 2) - - - 2- lwn‘" = n! 34. f(x) = 5x14 : (59: — 1)‘1 :> Hm) : —1(5m— 1)‘2 .5 : f"(a:)=(~1)(~2)(5:c—1)_3-52 => f’"<x> : (—1>(—2)<v3)(5w — 1V4 - 53 : => Mo) = <—1>”n!5”(5m — 1W“) 35. f(.7:) : e21 :> f’(\$) : 2627: => f"(m) = 2 - 262m 2 22623” => fli/(\$) : 22 . 262a: : 236210 :> ‘ _ ' : f(n)(\$) : 2n62z 36. f(m):\/E=a:1/2 : “saggy“? :» f"<x>=%<—%)m-3/2 2 f’"(x>=%<e%)<—%>w-5/2 : Wm) = %(~%)<—3)(—2>m-"2 _ —1 '234'5w~7/2 : Mm):%<~%>(—%><—2)(—§)w*9/2~1 32': 7m 9/2 a a, f("’(ar)=%(*%)(-%) (é—n+1)m~(2n~1)/2:(_1)n_11 3 5 ----- (2n—3)f(2n71)/2 37. f(\$) : 1/(3333) : %m*3 :> flcr) : %(_3)\$—4 :> f”(\$) : %(~3)(_4)\$75 : f’”(w) = §(—3)(—4)(—5)m*6 : :> 0» 4— _ ..._ —<n+s>¥<~l>"-3~4-5~~-<n+2>.2 M f (1}) — 3( 3)( 4) l (TL-I- 2)I\$ 3\$n+3 2 _ 6w"+3 38. Dsinm = c031: => D2 sina: : —sina: :> D3 sinzL' = — 00816 : D4sin1: = sincc. The derivatives of sina: occur in a cycle of four. Since 74 2 4(18) + 2, we have D74 sinx = D2 sins: = — sinx. 39. Let f(a:) : coszc. Then Df(2m) : 2f’(2\$), D2f(2:E) = 22f”(2:c), D3f(2:c) : 23f"’(2.r), ..., D(")f(2ac) = 2”f(n) (2m) Since the derivatives of cos 3: occur in a cycle of four, and since 103 2 4(25) + 3, we have f<103)(m) : f(3)(m) : sing: and D103 cos 23: = 21°3f(1°3)(2m) = 2103 sin 2m. ...
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