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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(y1)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 mewkm, 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 kin6M 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 ’ jiMf 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 nonzero 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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