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MATH 2204 Sample Exam 1 Instructor: Phanuel Mariano
Tame:
Instructions:
• All answers must be written clearly.
• You may use a calculator (TI-84 or below), but you must show all your work in order to receive credit. This includes any multiple choice questions! No credit will be given to any problem unless work is shown.
• Be sure to erase or cross out any work that you do not want graded.
• If two answers are circled in the multiple choice, then zero credit is given.
• If you need extra space, you may use the back sides of the exam pages (if you do, please write me a note so that I know where to look).
• Any cheating will result in an immediate F in the course.
• Partial credit will be given to open ended problems.
• The ACTUAL EXAM will not be this long! This is just to give you an idea of the type and style of questions to be asked
Question: 1 2 3 4 5 6 7 8 9 Total
Points: 15 8 8 23 10 15 10 5 10 104
Score:
ii,olt ~✓ .....,,)
l. Match the following slope fields:
\ \ \ \ \ \ \ \ \ \ \ \
\ \ \
\ \ \ \ \
\ \ \ \ \
\ \ \ \ \
(I) \ \ \ \ \
r r I I 1 r ! I r r 1 I 1 ' r I
1 ' I I r r I I 1 1 1 I 1 r 1 I
(III) 1 r I I
y
y
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \. \
\ \ \ \
\ \ \ \ \ \ \ \
- · - /
_ ___./
___./
___. /
___./
(a) t=(y-2)(y+ l )
I ✓ (b) ~ = (t - 1)2
--.--
i l f
(c) !flt = y(t + l )(y - 1)
Tr (d) 1t=(t-l)(t+5)
1/ov,, P
(e) t = - (y2 + 1) -j_
\ \ \ \ \ ".
\. "" \ \ "'8; \ \ \ I (II) \ \ \ , /
y
I 1 ! 1 / I I I
/ I I I 1 , r r r 1 r r r r r r r r r r
-· --. -----:: - ---·- -------r' - -:._---=- ---- _:, ~ \\\ \\\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
(15)
)--: . \\ \\\:-,,. --~ - ; I I - I ; ; - 7.....---> r r r r r r r r 1
(1v) r r r r r r r r r
Page 1 of 10
V ;;-1 , lj::: X d:£ -i-ii <--~\ -~n<::.___v.A,--~
2. If y = y(x) is a solution of v (8)
I Y X y = - + -, X > 0
X y
then which implicit equation must it satisfy? {: G)1t t)
A . y = x ( ~ + C) x2
B. lny = - + x + C 2
C. y2 = x 2 ln x + C
D. y = x ln x + Cx
~ 2x2 lnx + C~
\
- Vt _l - / V
?<& - _L - v f"('
\vJv,j~dK 2.,
~:: l1-,K + C J_
3. Find the implicit solution of t he initial value problem:
(6x2 y2 + 4ex - 2ysin2x) + (4x3y + cos2x) dy = O, ~ L.:---.----J dx
y(O) = 1.
A. x 3 y2 + y cos 2x + 4yex = O V B. 2x3y2 - ycos 2x + 4ex = 3 di"'(.(( if t1r-ad· ( C. x3 y2 + y2 cos 2x + 4ex = 5
{1;[__ 2x3y2 + y cos 2x + 4ex = ~ E. 2x3y2 + y cos 2x - 4yex = - 4
M r.:. 12 ~ \ - ,..o - J. r;.,, 2r l( ~-c~ 11 I(~_, l s,'L J.-r
(8)
✓
~ ~ S ( '-l x} + t D) c hJ) J 1 -+- 9 cJ :::.J ~ "y ?__ + y t v > c 2 ,0 + <J c1) ~
~ [?K\'+Lf/ .;,y10,(u) ;;- C] ~ . ~
Page 2 of 10 [C-:: r)
® 4. Consider the following differential equation dy = (y + 2)2(y - l )(y - 4).
dt (a) Find the equilibrium solutions for this differential equation.
® /CJ) y UlA >+<i~ lR
~}q~ l f
- )., >e~,,1-<~l{
(5)
I
(b) Draw a Phase Line for t his differential equation. Classify the equilibrium solutions. (10)
(c) Sketch a general graph of possible heh tion. Iny e the equilibrium solutio
of solut ions to this differential equayour graph.
(8)
----~ =-----------) f;
L--_ _J_----=--~:::::::======~-=--Page 3 of 10
5. Consider the following initial value problem: (10)
dy ln 120 - 4tl dt - y = t2 - 9 ' y(4) = - 3-
For what interval can we can guarantee a unique solution exists?
~ L1:: ~(
1 c~J .,_ l 11 l ) 0 - '1-t) l- -)
k -~
Jo - '1+ fO =>) { 7 S
t:1
- ~;£ b :::,':) t ~ J:.,)
(9 &i<'v<v0:0?_)
s;
c,~ Y Q ;r:cVJ] 14, b/ 'j<jJJ, I- 1 , -f P✓v, /
I) ti @
1 v/h c,.r+e,;.1.,t1 1 ~ ;:, lt --t~+ ~ VC{ ✓<,, .,,,, f (? Q) {J~l- tvP
tQ J lief rt~uif@.
Page 4 of 10
(15)
10
'\ C
.,
5. Suppose a 30-gallon tank contains 15-gallons of pure water. At time t = 0, we do two things:
• we start pouring in 1 pound/gallon of sugar at a rate of 3 gallons/minute; and• we open a drain at the bottom of the tank so that the mixture (sugar & water)
pours out at a rate of 2 gallons/minute.
Write down an initial value problem modeling the amount y(t) of sugar at time t, and solve it to find y(t). How much sugar is in the tank when the tank is full?
�y Pctf-e l1' F' Ctf f vv1.,-
d t _y_ .1 J � ,,) lhl j1§�li'1
-·--
t S-t t ____,..�u I yvt IV\
Hf qy yet Ill) ( 1 CJ e; e I
( t):: A)'\-1.YVV\i of J y_ i 3+
...,----- y )V� L.tV cti --li Y"\ f ·{
i__}d b * I
'r ( u) :: u
1 ln} 15-i-tl
---- e j
lntl-C+-tJ �
: ll; +t)1.
\v t\ PV\ y [ 0 __, l)
y l -t. -�
.....
L b c-0
3 l )
' _L [ ) � l-i) 1 ( �) J {
;V!(-l-)
) 5 ll T ..,-(/ j --\-
(i r � t) ·1
(1 -; + t) '3 +
( I J-+ {) J.
(!5+'t)')- )�7
(1�+),l
l
---- - ------------- - - -- - - -- -
:: - 1 t; > ,;
Page 5 of 12
t c 1 -i;) � ;. � , 2 r ; b
S -t I
f'\,, I V\
7. Find the general solution to the following linear differential equations
(a) dy - t2 dt + 2ty = 4e .
l),, ( ft,_"'
y ~
,.__ __________ ------.., Iv=- Lf 't -4-
L .-- --------- ·-
Page 6 of 10
(c) dy 2 1 dt + ty = t - 1 + t'
/\II LI I 4 IV' I y A { {� () l/ I
----
�[�/yjcH: Jc+ i-* . t) J t
fl
t LI
-t ') -t � t ..__ -+ )/ -
·+-
--
½ ) l
-
) y
'�
�-J - -t - t \ - ---
L y J ..2
Page 10 of 12
S,Ji fa(-l}:.. e
)-¼/~ ~ ('
l.l ~ -l - e ~ ?I
'( ;
dy 2 1 - + - y=t - 1 + - . dt t t
-'- l + C J_.
Page 7 of 10
8. Find t he general solution to the following separable diffcreutial equation
(a)
(b)
- l. u -- cl
-+- C
-_t_ ~ --- 4,- C 1-f
) (-{ d f
dy t dt = y (l + t2)3
j y =- j_ _J_ ·---
]_ ( 1-t .f '))._
age 9 of 10
(5)
C
+(
9. Consider the following initial value problem, (10)
!~ = y2 + t3 l y(O) = l.
By hand , use Euler 's method with h = 1 of Y2. Use the table to record t step. Show all wo:;
k tk Yk f (tk, Yk)
0 0 l 1\0~: )IJ
1 y \; Y1:1 t-.f'(l, .r) ~ d-
'}
I -t- l
:, 1 t i. / ;;.- tf] - :l_
2 J_ y). :: 11 .... .fr+, 1',) ~
::. :t- + r ·/ ::. r
Page 10 of 10
<
10. Solve the following IVP:
y" - y' - 20y = 0, y(0) = 3, y'(0) = -3.
l...
f - v --JO ::;,O
( v~~)C~+tj)
-Iii ~~
f c.,.rf , .. ,,,., ~
}!.,
-1 :. ) ( ,-(l) -'1 (1...
-1-:. l) - rc.l... -Lt l, _..l ~ l > - 'f ( l..
'¥t :.. If
fo iV.-1.!...f_J-,~! -::-:------
f l.t):::, , {, + J- e,, " Page 10 of 10
