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22 - #1 Section 2.2 page 47 x2 y'= y 512 3i dx y Ex 2 V2.1...

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Unformatted text preview: #1, Section 2.2, page 47 x2 y'= - y 512.: _ 3i dx y Ex. 2 V2 .1, dx x'jjly y W4 yrmly ydy = xzdx J-ydy = )6de 2 3 Z“ x— + C 2 3 j, #3, Section 2.2, page 47 I v : y'+y2 sin x : 0 dy 2 . ”*2"- Slnx dx y l 766/ = —sin xdx y i Jédy = — Isin xdx Iygzdy = — J. sin xdx —l y_1 2—(—cosx)+C 1 ‘V ——- =cosx+C y 1 ““M LL22)! ‘2 XSQLI'M'g’J“ ’OJS':1"2 :3? ercauclM dfziJ“ 2;) j $9.62)!ch = Jw’x cw 3L fonflly) 3 11“ (1+ 1“ Sikh) +C’ 'tantly) 21‘ 1+ é Sl‘h2X+ 2C, gummy) : 2m sinzm 4c} 1......) Id t“: be anofher arbi‘lmxy constant (T, amine/r1 2 tany) ~ I» sinm C W H §2.2 77. .42.- we o\x " we! -X went»: (14‘va My aficiyz- “WE‘XMX iyz‘fe)’ z: i1“-(~a}€‘x+C’ §71+e7 == éxl+€“"+c’ 72267: 12+2e‘x+2c’ 71~x1+2et 26"": 2c} L.__-...._J ldi’cbeC am: ylfill‘f Meier“): C VuflMi X30 OVM‘ 7/: “é ink: ____L.... ”J. : O’OWC B (1:6 =2? --§—= X~X2+L I yz‘m—mv-W x~x‘+e #11, Section 2.2, page 48 xdx + ye‘xdy : O xdx = —ye‘xdy xexdx = —ydy Ixexdx : —_[ydy 2 xex ~ex = —y—+C 2 12 060 ~60 : —— + C T 2 ~12 «1+C 2 c = —i 2 y2 1 Here: Jxexdx : km-w' f3 dx xex — Jexdx :xe“ —ex Initial condition: y(0) =1e—~>{ §2J. .L #I‘f‘ ‘7’: 7a/3(I+x‘) , ‘/‘°>=J W 3 d 6% 3—; = 135W?) -3 - 1 7 A] ~ HX‘ d1 -34 :. j de f7 7 Jm‘ "‘9 ~§Z= (MW-c " PuftUVlj x:o and ya! {M70 r1 * ‘1‘ :- 2 (HO) +C ”-21- : Hc, \<>< a: n #19, Section 2.2, page 48 sin 2xdx + cos3ydy = O COS 3ydy = “sin Zxdx 1 1 -sin3y=—cos2x+C 3 2 1sin 3-1 =~l~cos 2-3E +C 3 3 2 2 lsimr = j-<:057z+C 3 2 O: -—1+C l . 3 C II 1 5 1 3 Solution: lsin3y=l0052x+C 3 2 lsin3y=lcos2x+l 3 2 2 lsin3 '~l(1+0052x) 3 )22 l . ~sm3yzcoszx ’3 3 sin3y : 30032 x q _ sin"1 (3 cos2 x) —> rejected!!! 3)» — 7r — sin‘1 (3 cos2 x) 3y : 7r _ sin“ (3 cos2 x) 1 y = 3 (7r — sin—1(3 cos2 x)) #20, Section 2.2, page 48 1 y2 (1 —— x2)E dy = arcsin xdx sin‘1 x yzdy: 1 2ai’x —x {2%(sin—‘xfw D 3 1 « ~1 2 —§—:E(sm O) +C l21-0+C 3 2 CI; 3 3 l; r. %(sin‘1x)2 + C y3 1 -»l 2 1 ——sm x +— 3 2'( ) 3 3 3 ‘ —~J 2 y =~2~(sm x) +1 y = 3\/3 (sin‘l x)2 +1 2 Integration by u—substitution c d . ,_l 1 Since—Sin x = :— , we may let dx 1“ x“ u = sin‘1 x . Then: fl .2 1 dx l—x2 du : 1 dx l—x2 sin”l x 1 dx : sin—1x a dx IVE—x2 I \[1 _x2 2 Iudu :luZ 2 = —1—(sini1 v)2 + C 2 Initial condition: 32(0) 219—»? y :1 #31, section 2.2, page 50 £12m x2 +xy+y2 dx x2 dy x2 xy )2 “=7+7+—3 x xi 21+ v2 (— separable dx 1 (fizzldx 1+1;2 x tan” v = In x + C tan"1 2: = In x + C x IMPORTANT!!! A x yzvx dy d \ -——:— vx dx dx< J dx dx €3=v+xéfi dx dx 10 #33, section 2.2, page 50 fl“ 4y—3x<——”—— dx ' 2x—y «if—m y x 4»~—3~ dy: x x ch £_1 x ‘c y 4__3 fl: x dx 2_Z x y dv 4Z—3 v+x-~= “ cbc _X x dv 4v—3 v+x——: dx Z—V dv 4v—3 x—w: ~v (iv 2—v xfl_4v—3_v(2—fl (Ix 2—v 2—v xfl_4v—3—2v+v2 cbc 2~v xfl_—3+2v+v2 dx 2—v 2~v 1 —2————— (IV : — dx <— separable v +2\)—3 x ' ' 5 i_1__; 1 dvzldx 4v—1 4v+3 x llnlv ~ 1’ ~21n‘v ~-- 3} = lnkx} + C 4 4 11n y ’1 —~~—3 = ln‘x} + C x X—llmiln x v = 1’— x y = we > 1’ g 9—} 2 —d—(vx) dx dx i dy dx dv ; ~— : v— + x— dx dx dx 99=v+x93 dx x Partial Fractions 7 _;&2;_:? v2 + 2v — 3 U 2 ‘V _ i kg. (v—l)(v+3) v—l v+3 2-—v:A(v+3)+B(V—1) 12:1 ma2«1:A(1+3)-+B(1*1)—>A::li v:—3 2~v _ A y B (ya—1mg) "v—l ‘ v+3 _1_L 5 1 “Zv«1“Zv+3 —92—(—~3)=A(w3+3)+B(—3—]) —>B:—4 5 ll OKX ” 1.15 é. ~Xu 1 1 fl 3 I» 3 (K) Lei V21 Ax 2(‘1 x x) .. y~V 0W I~3v’ ML... .. d V o\x 2v 2&- ‘“ fihx) 0W 2 [’3" "-1" ix... ELK X31 2v ”V K95“? X ax fig - 1*3v-2v3 Av X (AX 2v ~ V+X 33; £11 :: i~ Sv‘ X0\X 2v 2V .. l’5v‘ 0W .. FXL‘AX _____12v =1 'Lc} 2v I~5V dV X X “—9 I 1&5“ 4V, ’th w: ~-5':MI~5V|:, lanHC’ :3 2V ' 0‘ $1 2 _ W'W W ~HMI~H3¥H~ wmc’ - ivy—Lay ”2 TM 64 om acceptable QM, 5 W w LAS— 1M, :3 *”§‘ 23% l “51/"! -%Jm\!~5(%)’} = 1mm ’0“ ‘ ‘"5 (391-; : |VLIM+ Cl. --» Tulle QXPOWlequ'd Pow 51*5(%)‘l ~'= We} “find 1 C 1"” £4! :Clll'5 ‘mtigli‘ :C’lxrf D94)” = C m“? “(In H557"! 2 cm“ X 3—31“ tu- ~xt fl : 3%-: lei v dx .1 2>< dv ~ 3v1~| “W “37““ o\V ... 1 LOW - 3%: “v Xo‘V - 3v~l~2v‘ 33? 2V JV :1: Va! x35? 2v fidv - ‘3‘} dx Vi”. X é‘“ V"! dV BMW—1| :‘XMIXHC’ 2-" 3!...9‘31 2 w 2v WW! = Mme == Wm /\ J: ble MSW ‘1 fivlwl c ox. 0" m“ °"’ :: m I M Or alimfiiwly, ‘tqke Wonayxifmfl, H3654} 2 m g: 1 z C W... ~ we ...
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