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test1AB_ans - TEST 1 Version A NO CALCULATORS r MAP 2302...

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Unformatted text preview: TEST 1 Version A (07/15/11 NO CALCULATORS , r; MAP 2302 ) Name, SOZU/ /0U$ Summer B 2011 3rd period Section 4787 Please use the blank space and the other side of the sheet to write your solutions, numbering them clearly. 1. (2 pts). Determine whether the equation (:32 sin :13 + 4y)dx + zdy = 0 is exact, or not. If it is, then solve it. If it is not, then do not solve it, but check whether the function 1 (1M. _ {W N (9y 83: is a function of a: only, or the function i<3_N_?_A£> is a function of y only, and find an integrating factor. Mal/a) ‘2 XZS'A‘MX +47 MW, 3 ) t X 9 ’9 fl‘ «£4,544 93- , 1 ’MWUf" ((745, UK 2. (2 pts) Determine whether the equation (eaC siny — $3)dx + (ex cosy + y2)dy = 0 is exact, or not. If it is, then solve it. If it is not, then do not solve it, and find an integrating th 'P bl 1. x ‘ = - ' resamewayasm’ro em SW06“ Fag)AC/WW 3. (2 pts) Determine the order of each of the following equation s and classify them as linear or non—linear. 1 dsy 2 dy g . a) $+x (fl +ys1n33—0 d4y d2y - 4 - 2 __ b) 8111(1) )% +sm(m )Eg—i‘i‘y—SE / afi 0.) 3/106 Mata) .W’JZLMd/‘L {Mu—AMMW {7M 4) H4 mm, 124W“ 4. Water containing a salt with concentration 0.05kg.L_1 is flowing into a very large tank at the rate of 20L.min_1. The tank initially contained 100L of water with 1kg of salt dissolved in it, and the solution is stirred to keep it mixed. The water flows out of the tank at the rate 10L.min“1. (a) (2 pts) Set up the initial value problem describing the above situation, i.e. write the differential equation with the unknown function being the mass of salt in the tank as a function of time, as well as the initial condition. SW £24 6m Wm 5(Wt’ZJ/ué. "1* (b) (2 pts) Solve the initial value problem you have set up, i.e. find the mass of salt dissolved in the tank as an explicit function of time (before the tank overflows). SW 52A {/14 WW Bi W 6W 5. (a) (2 pts) Does the Theorem on the Existence and Uniqueness of a Solution of the Initial Value Problem y’ : f (x, y), y(x0) : yo imply that the initial value problem wlm y’=3(y-1)i y(O)=1 has a unique solution? Justify your answer. 2. “QZVW/ i./afzd76('g):3(g~l)i ,M*&(“S‘WWWL gi- =2(3~;)“é ~MWMM mew-km, 72/, n [a fluz Wwi 55E! (b) (2 pts) Find all the solutions of the. initial value problem in part (a). “7’“ along JJL/a—Z— (on 774%? canning I ' 3km C 2 0. (”0‘03 =XfC kin-6M 2%? WM gm: (+(X+C)3 fwwé'mfféfi/ W M60424 tan {mi-(«1 m 6. Suppose that the equation describes the quantity y as a function of time t. (a) (2 pts) Without solving the equation find the equilibrium solutions, and identify each non—zero equilibrium solution as a sink, a source, or a node, using a sketch of the solution curves. (b) (2 pts) Can the value of y(t) ever exceed 3, if the initial value was y(0) = 1? Justify your answer. SW. 614 51% mm [3,. See €W~ 7. (a) (2 pts) Identify the following equation as homogeneous or Bernoulli, then use the appropriate substitution to transform the equation into a separable equation or a linear equation (with the new variables). (b) (2 pts) Solve the transformed equation and write also the solution of the original equation. . SLFMm/j .' ”(j f a K Lint/V 8. (a) (2 pts) Identify the following equation as homogeneous or Bernoulli, then use the appropriate substitution to transform the equation into a separable equation or a linear equation (With the new variables). x a: _ 1323/2 4 i F2 «» 2A a. Liam 2%” 3M‘Wm "’ Whjfljé) I 9. - Bonus (2 pts) Verify that the relation may + y : C, C is a constant, is an implicit solution of the equation TEST 1 Version B (07/15/11) NO CALCULATORS , . m A MAP 2302 Name: SOLD” IO/i/é Summer B 2011 3rd period Section 4787 Please use the blank space and the other side of the sheet to write your solutions, numbering them clearly. 1. (2 pts) Determine the order of each of the following equation 5 and classify them as linear or non—linear. _d__3y dy+ a) 3—553 +332 gal—30+ y 2 0 4 2 b) sin(a:4)d—y + tan(:r 2)d— y (11354 h —2$+y=sinm CUE “(9410601 m Luau; («:Lfm Zgwmfifibm x6“ 59;) b) 415/; OWL/€11) Jen/weal < 2. Water containing a salt with concentration 0.051691‘1 is flowing into a very large tank at the rate of 20L.mm‘1. The tank initially contained 100L of water with 1kg of salt dissolved in it, and the solution is stirred to keep it mixed. The water flows out of the tank at the rate 10L.mz’n_1. (a) (2 pts) Set up the initial value problem describing the above situation, i.e. write the differential equation with the unknown function being the mass of salt in the tank as a function of time, as well as the initial condition. Fina/fan I» .015; 2. (0 0:, >620) ~ JQ,L_._ g x /6’0+/0t ’ "‘ "' W #ZEW / ) afaée W Miriacfg W 07%: flame Wham {(0) / “WW (b) (2 pts) Solve the initial value problem you have set up, i.e. find the mass of salt dissolved in the tank as an explicit function of time (before the tank overflows). 14”,; Q/ZCMIl/i efuaj’c'm‘ M X 1. / /+/—— f M 42‘ MM ‘ (kg) avg 10% /fl+f grim/221417, 5MM -, K:(f>‘= //fl+f) //0+{) HM‘I’C} 1- /0+6 3. (2 pts). Determine whether the equation (yey + x2)dy + rydx = O is exact, or not. If it is, then solve it. If it is not, then do not solve it, but check whether the function 1 flJLN N 8y (9:6 is a function of x only, or the function i LN _ 5_M M <93: 33,! is a function of 3/ only, and find an integrating factor. Win/0)“? Laid :JE.=_L Wyth'W” M 9/ x a) gfizx igélx Scrfi(a)=f€f%%:/&/,§MMW13 (it’s ok IF IT’S TAKE” (Jakggy' 4. (2 pts) Determine whether the equation $2 '3? (xsiny—x3)d$+ (Ecosy+y3)dy= O is exact, or not. If it is, then solve it. If it is not, then do not solve it, and find an integrating the same way as in Problem 3. f N (x ) ~ xum r~2<3 gmim' Al ‘ a ’3 F {x ) = C W / X?’ 7) l 3 I W 3%? *7 «a + ~ m x ’3 ) 2 f ‘ Q 006 +5 “2 X000 a‘fi/(Coo 9 i g 2? item + kw, >41: a/fixm/f / K ( ’2 (j a j 9 5. (a) (2 pts) Identify the following equation as homogeneous or Bernoulli, then use the appropriate substitution to transform the equation into a separable equation or a linear equation (with the new variables). __ 4x2 + 2y2 _ my , x>0,y>0. (b) (2 pts) Solve the transformed equation and write also the solution of the original equation. a Sr M3 ’ ji-Mf fx 4/. flair—- : Ax+C B 6. (a) (2 pts) Does the Theorem on the Existence and Uniqueness of a Solution of the Initial Value Problem y’ = f(a:, y), y(x0) 2 yo imply that the initial value problem CHI»;> y’=5(y—2), y(0)=2 has a unique solution? Justify your answer. i a}; W 7’=(f&,;)= 5/9172)? . ([6973) £er 2% ‘1 WW . ' : (Mguz) F ~MW‘MM M‘W'Mfm 9‘2, ’4” 93 iifixzmm/vz1‘s.~ (b) (2 pts) Find all the solutions of the initial value problem in part (a). describes the quantity y as a function of time t. (a) (2 pts) Without solving the equation find the equilibrium solutions, and identify each non-zero equilibrium solution as a sink, a source, or a node, using a sketch of the solution curves. 1‘ :02 (MM y [a/O).>_2 (b) (2 pts) Can the value of y(t) ever exceed 3, if the initial value was y(0) = 1? Justify our answer. gm \ Race. ‘flo’ umotg , 36%,)5M1m WW4W fir wwd Mu: quif/égfium J‘M’M 966-) =°Z m ,% Wmicds sew mm flows/U40? M4 WW Limb/1m 9 1:155 {1/12 8. (a) (2 pts) Identify the following equation as homogeneous or Bernoulli, then use the appropriate substitution to transform the equation into a separable equation or a linear equation (with the new variables). (b) (2 pts) Solve the transformed equation and write also all the solutions of the original equation.‘ i 7/4 3 d WM 2'7“" ("51/4 . 1. WJ yMfl‘mflh/fi/JQXXMX 1‘ C} 9. - Bonus (2 pts) Verify that the relation xe2y + y = C', C is a constant, is an implicit solution of the equation { 3mm 12(0th (:21 ‘ M4/7z1c9’14 c; €23 ”iii/:7) + 2’ = 0 ...
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