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```MATH 245: Differential Equations and Linear Algebra
SAMPLE MIDTERM
I
Fatl'2009
NAME
,
hNSv/ER ReY
NO{E: There a.re 5 problems on this midterm (totat of 6 pages). Use of
calculators to ctreck your work is permitted; however, in order to receive full
credit for any problem, you must show work leading to your answer. You
have 50 minutes to complete this test.
Problem Possible
points
1
20
2
20
3
20
4
20
5
20
Total
100
Score
I
MATH 245 - MIDTERM
Page 2
Problem 1. (20pts) At noon a ca,r starts from rest at point A and proceeds
at constant acceleration along a straight road toward point B. If the car
reaches B at 12:50 PM. with a velocity of 60 mph, what is the distance from
AtoB.
L-t
( d.,tt"^." +,--"lUt )
X(t) = Pos,l6^
I
M;\p4
cI
Giv.rn X(ta) -
O
Xt(rz1 =- o
tr ttzf) = 60
tt' =
F\I
(")rt6A,^L
r
x(rz{1
Xt=
of t
ga{
J
V"
= a (t- lr)
X =-
Itt = a- -
s,uc-e
I a tt- t'\" t d"
z-'-
e\'\\-cA a (rz\*- o '
t
6o = ^-
( E)
x!zl)
-t
:
c')
1'(rz)--o.
&c 72 *ip,S
8-)(E)25 r;lL>
MATH
245 _
MIDTERM I
Problem 2. (20pts) A pitcher of buttermilk initally at 25C is to be cooled
by setting it on the front porch, where the temperature is 0'C. Suppose that
the temperature of the buttermilk has dropped to 15"C after 20 min. When
will it be at 5'C?
-f:
bo"p"*"t^r"
\
, t'
Lw',r
- dt,: -ktfT-o)- -kT
dta
lr-
[rtt.*"ltL-.)
ln -LHg =) l^ T: -Lt
)r
...
e,
)
[:
C
e-bt_
T(") = 25
C: Z€
fzo)=
ti
15
= zS
e-
?sV
;\= E
Y-53
D;t *.-L
-tt";5
g'l t e,+t* S=ZS1*
*=s
r-3*
u--/
^zntE1")
.,)rZ
= Lf I
L* = JJ*(s)
=
ry
-!^
5
JnS - 'lq3
L = 2-/)J"5
€n^irr,(rr)
MATH 245,- MIDTEF.M
Problem 3. (20pts)
I
Solve the following
rg' +
3u
J'o F")u
'
,,
u(2)
:,
L^t
f UIAt )' -- 2x7
i
rys /
dx-= +t I
v = t + Z-+v
il 4r
Pace 4
initial value problem.
- 2x1,
=
,
(3
J*
Le = <u'' = xo
Z=+'-C
) ,l--f
II4\$TH245-MIPTERJvII ,,
.
P+ee5
(20pts) Appty Euler's method with a step size h : 0-1 to
approximate the solution to the following initial value problem on the interval
[0,.5].
Problem
4.
A':U-r-1,
Xn*,
tl=
)tt+(
0-0
t.o
0-I
t_o
'
g(0):1
Tr^, '),
t/ n .lt-
J^
0- z,1"1
L)
I
g^
ln
9_
, 117
1557
"881+7
(-a) - 7-o4
{-o t o-t (t- e-r -1)
.?1 +o-l(^11 *0-Z-1-):'1'1 -"0z-1 ='1L"1/
7.0+ o-t(
t 0-t (-ttT- -3- L)= -161 ,1ts1 * o.t (.1ts7--+-L)
.1[7
, -1js7"
= e, 691+1
vvrrr
o"t (-TG+()
- 033
I - .13fT
MAIH
245
-
MIDTERM
I
Pase 6
Problem 5. (20pts) Find the general solution to the following difierential
equation.
u"'+4u':5e2' *3r-
1
-c =o)lzi
(ft 't) : o
Jn^**U^r^", g-\,*'. 7^, @ ZA ,sM2*d
Yeo4c< l-
ftl^+'.-l-. +&
V\
t(
Yu
JT
Js
: +A? +
Yfrf
rr
t,
At* *
Bxor Cx
LB
+
+ +(A;"
=)
'tc:
=>
-(
* z6x,
c) :
t;. t3x*
)
/- t/,lr
l6A=5
gh=3
G--
'=
ZAt" * Zf) Xr-C
=
,,{ "'
Yfr
u
:,
B = yg/r
C = -t/rr
9nl,n
Ju,
= cr + tzL'?u
t Qstalx r
Et-r ?i-
17
```