27[Sanple Final Exam.
Score
N
Solve the following homogeneous differential equation.
1) 5y" - 3Y'+ 3Y = fl
A) y =
"nEnoX[ql
(@"'l:1
['i'*
,io*,
.
*:-
o "" #1]
c2 .o,
1)
*;l
C) y = g1s(3[01x * gre^,E7r0x
D) y = e(-3l1ox[c1
r.#,..
.,
"",
#.]
Find the solution to the following differential equation satisfying the given endpoint conditions.
e-1!L--\
We-xsin3x
(a
2)D2y*2Dy+10y=0wift Y(0)=0andy(n/5)
A)y=exsin3x
C) y = g-x cos 3x
D) y = e-x (C1 sin3x
Use the method of undetermined coefficients to solve the
3) y" * 4y' + 4y = 6e-X
A) y = (C1 + C2x)e-2x * 6vs-x
2)
+ C2 cos
3x)
differential equation.
3)
CQgl+c2ge-2x+6e-x
'
D)y=(C1 +C2x)e-2x- 6e-x
C)y=C1e-2x +C2&x+6e-x
Solve the given differential equation.
q
Y"r+
49Y =cos
7x
4)
A) y= c1 sin 7x + c2 cos z*
*fi^cos
B) y = c1 sin 7x + c2 cos 7x -
7x
fix *rrzx
r*.I*"^r:
CC"r*l.+c2cos
D) y = c1 sin 7x + c2 cos 7x +
fixsin
7x +
1,
fix
Use the method of undeterurined coefficients to solve the
5) y" - 25y = 35;r, x with Y(0) = 0 and y'(0) = Q.
olr=*-d*C) y = -
_
cos 7x
differential equation.
5)
x
osx
"-5,,-fr "or*
fr"s, * fr."-s, - fi
"o,
*
D) y=
--3
"-sx
-a
sin x
-*u**7r1-"-s* -S
'm,,
,/14
Use the method of variation of parameters to solve the
differential equation.
6)Y"*Y=sedx
6)
- 1 + cos x lnlsec x + tan xl
+
B) y= Ct cos x C2 sinx - secx sinx + sinx Inlcsc x + cotxl
C)y = Ct cos x + C2 sin x - sec x sin x + cos x ln I sec x + tan x
D\i =C1 cos x + C2sinx - L + sin x lnlsec x + tan xl
A) y = C1
cos x + C2sin x
I
Solve the problem.
7) A spring with a spring constant k of.175 newtons per meter is loaded with a 7-kilogram mass
and allowed to reach equilibrium. It is then stretched an additional0.S meter and released.
the equation of motion. Neglect friction.
B) y = o.g cos 15.5531
= 0.8 cos 5t
D) y = 0.8 sin 5t
C) Y = O.g sin 25t
8)FindthechargeQasafunctionoftimeinanRCLcircuitifR=105C),L=0,C=5,10-5F,
n
8)
and E = 5 V. Aszume that Q = 0 and I = 0 at t = 0 (when the switch is closed).
A)Q=5,10-6(1-"-0.2t)
C)Q=r.5,19-51t-e-t)
,--@= 2.5 x 1o-s (1 - "-o'29
D)e=u,1s-61t-e-g
(
9)FindthecurrentlasafunctionoftimeinanRLCcircuitifR=10O,L=0.025H,C=0.00014
AY{=7.27e-2001 s6
))
6991
I = 1.50e-240t sin 5961
e)
B) | = 1.27 e-2401 gss 6991
D) I = 1.20e-200t s6 5,691
initially contains 100 gal of brine in which 20 lb of salt are dissolved. A brine containing
Slblgal of salt runs into the tank at the rate of 5 gafmin. The mixture is kept uniform by stirring
10) A tank
10)
and flows out of the tank at the rate of 4 gafmin. Find the solution to the differential equation
modeling the mixing process if the variable y is the concentration of salt in the tank at time t.
- 1.8 x 1010
A)v=3-'J
(100+t)a
c)v=3*
B)v=3-
2'8x108(100 + 05
1'8x1019
(100 + $5
and which are unstable.
y' = (y - 3Xy - 5)3(y - 6,)
A) y = 3 is a stable equilibrium value and y = 5 and y = 6 areunstable equilibria.
= 5 is a stable equilibrium value and y = 3 and y = 5 are unstable equilibria.
,-3D
Cl I= 6 is a stableequilibriumvalue andy=5 andy =3 areunstableequilibria.
(
\
D) y= 5 is a semi-stableequilibriumandy =3 and y =6areunstable equilibria.
11)
11)
Solve the Bernoulli differential equation.
12)3x2y' -2xy =y-3
N
y2
=+1*+
Cx1U3
qt'=**311/3
12)
Identifythe signs
rg)* =f
of/
and
y".
-7
13)
A)
'
y'<0
y'>o
y'>o
y'<
y" <0
y" >o
y" <o
y"
y'>o
y'<o
y'<
y'ro
y" <o
y'lro
y" <o
y" >o
y'.o
y'
.o
y'<o
y'>o
y" ao
y" <o
y" <o
y"
y'>o
y'<o
y'<
y',o
yt'>o
y" <o
y" >o
o
o
o
ro
to
y" >o
0L
-1
Solve the following problem.
14)The system of equations
'S
=
OO*
-
3x2
-axy ana
S = 42y -
3yZ '2xy is a competition
model for certain species x & y of lemurs. Find and classify the nontrivial equilibrium point.
Al (12,6); it's a cmter.
C) (12,4); it's an unstable spiral.
j,)
it's a stable 2-tangent node.
a saddle point.
D) (72,4); it's
l4')