Diffi Common Finals

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FORM A

Math 2214 Common Part of Final Exam Spring 2004

Instruction: Please enter your NAME, ID NUMBER, FORM designation, and CRN

NUMBER on your op-scan sheet. The CRN NUMBER should be written in the upperright-hand box labeled Course. Do not include the course number. In the box labeled

Form, write the appropriate test form letter A. Darken the appropriate circles belowyour ID number and Form designation. Use a #2 pencil; machine grading may ignore

faintly marked circles.

Mark your answers to the test question in row 1-11 of the op-scan sheet. Youhave 1 hour to complete this part of the final exam. Your score on this part of the final

exam will be the number of correct answers. Please turn in your op-scan sheet and thequestion sheet at the end of this part of the final exam.

1. Determine (without solving the differential equation) the largest interval on which

the solution to the initial value problem

t+1 y +y = tan(t), y(0) =1.

is certain to exist:

(a) (-1,) (b) (-p /2, p/2) (c) (-1, p /2) (d) (0, p/2)

2. The general solution of

y + 2 y + y = 0

is

(a) y(t) =C1t+C2 et+ C

3t e

-t

(b) y(t) = C1 + C2 et+C

3t e

t

(c) y(t) =C1t+C2 et+C3 t e

t

(d) y(t) = C1 + C2 e-t+ C3 t e

-t


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3. Consider the initial value problem

y = 2t y2

+y, y(-1) = 2.

If we use Eulers method with step size 1 to calculate

y(1), we obtain

(a) -8 (b) 0 (c) -4 (d) 4

4. Let y(t) be the solution of

t y = -y + t, y(1) = 0.

Then y(2) equals:

(a) 2 (b) 0 (c) 3/2 (d) 3/4

5. A tank initially contains 30 gallons of water in which 10 pounds of salt is dissolved.Fresh water is entering the tank at a rate of 2 gallons per minute, and the well-stirred

mixture is drained from the tank at the same rate. Let Q(t) denote the amount of saltin the tank at time t. Then Q(t) is a solution of the initial value problem:

(a) Q = -Q

15, Q(0) =10

(b) Q = -Q

15, Q(0) = 30

(c) Q = 2-

Q

15 , Q(0) =10

(d) Q = 2-Q

30, Q(0) =10


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6. The general solution to the differential equation

y =e-t

y

is:

(a) y(t) = C- 2e-t

(b) y(t) = C+ 2e-t

(c) y(t) = 2 e-t+C

(d) y(t) = e-t lny + C (implicit form)

7. The differential equation

y + y -2y = e3t

can be converted to which first order system:

(a)x1(t)

x2(t)

=

0 1

2 1

x1(t)

x2(t)

+

e3t

0

(b)x1(t)

x2(t)

=

0 1

2 -1

x1(t)

x2 (t)

+

0

e3t

(c) x1(t)x

2(t)

= 0 1

2 -1

x1(

t

)x2 (t)

+

e3t

0

(d)x1(t)

x2(t)

=

0 1

2 1

x1(t)

x2(t)

+

0

e3t


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sin(t) y + ety =1 is

(a) nonlinear (b) homogeneous (c) separable (d) linear

9. The general solution of

x1(t)

x2

(t

)

=

0 2

-2 4

x1(t)

x2 (t)

is:

(a)x

1(t)

x2(t)

= c1e

2t1

1

+ c2 e

-2t1

-1

(b)x

1(t)

x2(t)

= c1e

2t1

1

+ c2 e

2t1

-1

+ t

0

1/2

(c)x

1(t)

x2(t)

= c1e

2t1

1

+ c2 e

2tt

1

1

+

0

1/2

(d)x1(t)

x2(t)

= c1

-sin(2t)

-cos(2t)

+ c2

cos(2t)

-sin(2t)


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y + y = 5 t+ 3et

has the general solution:

(a) y(t) = C1 + C2 e-t-5 t+

5

2t2+ 3 t e

t

(b) y(t) =C1 +C2 e-t+ 5 t+

3

2e

t

(c) y(t) =C1 +C2 e- t- 5 t+

5

2t2+3

2e

t

(d) y(t) =C1 +C2 e-t+5

2t2+ 3 e

t

11. Consider the differential equation

t2

y + t y -y = t2.

The corresponding homogeneous equation has the solutions y1(t) = t and y2(t) = t-1.

The general solution of the given nonhomogeneous differential equation is

(a) y(t) = c1 t+ c2 t-1

(b) y(t) = c1t+ c

2t-1+

t2

3

(c) y(t) = c1t+ c

2t-1+ t

2

(d) y(t) = c1 t+ c2 t-1+ c3 t

2


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FORM A

Math 2214 Common Part of Final Exam Fall 2003

Instruction: Please enter your NAME, ID NUMBER, FORM designation, and CRN

NUMBER on your op-scan sheet. The CRN NUMBER should be written in the upper

right-hand box labeled Course. Do not include the course number. In the box labeledForm, write the appropriate test form letter A. Darken the appropriate circles below

your ID number and Form designation. Use a #2 pencil; machine grading may ignore

faintly marked circles.

Mark your answers to the test question in row 1-12 of the op-scan sheet. You

have 1 hour to complete this part of the final exam. Your score on this part of the finalexam will be the number of correct answers. Please turn in your op-scan sheet and the

question sheet at the end of this part of the final exam.

1. Consider the initial value problem tan( ) ln( ) , (2) 1 y t y t y + = = . Find the largest

interval in which a solution is guaranteed to exist.

(a) ( (b) ( (c) (1,2) 0,1) / 2,3 / 2) (d) (1,3 / 2)

2. Let A be a constant matrix such that(2 2)4 8

1 2A

=

and1 2

2 4A

=

. One

possible solution of the system A =y y is:

(a) (b) (c)2 2

2

4

2

t t

t

e e

e e

+

+ 2t t

2 2

2 2

2 12

4 3

t t

t

e e

e e

+

+

2

22

t

t

e

e

(d)2 2

2 2

2 8

4 2

t t

t t

e te

e te

3. Let solve the initial value problem( )y t 2 , (0) 1 yy t y = = . What is the value ?(2)y

(a) y = (b) y (c) y(2) 3 (2) 5= 2(2) e= (d) y e 2(2) 2=

4. A tank initially contains 1000 gal of water in which is dissolved 20 lb of salt. A valve

is opened at time t= 0 and water containing 0.2 lb of salt per gallon flows into thetank at a rate of 5 gal/min. The mixture in the tank is well stirred and drains from the

tank at the same rate of 5 gal/min. Determine the time t, in minutes, that the tankcontains 180 lb of salt.

(a) t e (b) t e (c) t180/200= 200/180= 200ln9= (d) t ln(180/ 200)=


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5. Let solve the initial value problem( )y t 24 6 , (1)ty y t y 1+ = =

e

. What is ?(2)y

(a) y = (b) y (c) y e (d)(2) 4 2(2) = 2(2) 2= (2) ln 2y =

6. Suppose you decide to approximate the solution of 2 24 , (1) 4ty y t y + = = using

Eulers method with step size h 1/ 2= . Your approximate value for is:(2)y

(a)5

(b)6

7 (c)

6

5

3 (d)

1

3

7. Let A be a constant matrix with real entries such that(2 2)1 1

(3 2 )1 1

i iA i

+ + = +

.

The solution of5

, (0)2

A

= =

y y y is:

(a) e (b) e35cos 2 5sin 2

2cos 2 3sin 2

tt t

t t

+ +

3

5cos 2 sin 2

2cos 2 3sin 2

tt t

t t

+ +

(c) e (d) e35cos 2 5sin 2

2cos 2 sin 2

tt t

t t

+

35cos 2 sin 2

2cos 2 sin 2

tt t

t t

+

8. The solution of the following initial value problem

'''' 2 ''' 3 '' 0, y y y+ = (0) 1, '(0) 1, ''(0) 0, '''(0) 0 y y y y= = = =

is

(a) 1 2 (b) (c) 13tt e e+ t t32 te e t (d) 3(1/ 6) (1/12) (3 / 4)t te e+ +

9. One solution of the 2nd

order linear differential equation

(2 '' 2 ' 2 0,t y ty y+ = 0t ),

is . Which of the following is the general solution?1y = t

(a) (b)31 2y c t c t = + 11 2y c t c t

= + (c) 1 2 y c t c t 2= + (d)2

1 2y c t c t = +

2


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10. Given the 2nd

order nonhomogeneous differential equation

3

t

+

+

+

,2'' 2 ' 5 sin(2 ) 2 4t t y y y e t te + + = +

the suitable form for a particular solution Y t , using the method of undetermined

coefficients, is

( )

(a) 2sin2t t Ae t Bte Ct + +

(b) 2cos 2 sin 2 ( )t t t Ae t Be t C Dt e Et Ft G + + + + +

(c) 2cos 2 sin 2 ( )t t t Ate t Bte t C Dt e Et Ft G + + + + +

(d) 2cos 2 sin 2 ( ) ( )t t t Ae t Be t t C Dt e Et Ft G + + + + +

11. The general solution of the 2

ndorder homogeneous equation

2 '' 2 0t y y =

is . A particular solution Y t for the corresponding nonhomogeneous21 2 y c t c t = + 1 ( )

equation

2 2'' 2 2t y y t =

is

(a) Y t 2 2( ) ( 2 / 3) ln | | (2 / 9)t t t= +

(b) Y t 4( ) ( 1/ 5)t=

(c) Y t 2( ) 4t=

(d) Y t 2( ) (1/ 3) (1/ 8) (1/ 6)t t= +

12. Given an initial value problem

,9 '' 0y y = (0) 6, '(0)y y= = ,

find so that the solution approaches zero as t . +

(a) = (b) = (c)1 2 2 = (d) 1/ 2 =


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FORM A

Math 2214 Common Part of Final Exam Spring 2003

Instructions: Please enter your NAME, ID NUMBER, FORM designation, and CRN

NUMBER on your op-scan sheet. The crn number should be written in the upperrighthand box labeled Course. Do not include the course number. In the box labeled

Form, write the appropriate test form letter A. Darken the appropriate circles below

your ID number and Form designation. Use a #2 pencil; machine grading may ignore

faintly marked circles. Mark your answers to the test question in row 1-12 of the op-

scan sheet. You have 1hour to complete this part of the final exam. Your score on this

part of the final examwill be the number of correct answers. Please turn in your op-scansheet and the questionsheet at the end of this part of the final exam.


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Form AMath 2214 Common Part of Final Exam May 6, 2002

Instructions: Please enter your NAME, ID NUMBER, Form designation, and INDEX NUMBER on

your op-scan sheet. The index number should be written in the upper right-hand box labeled Course.

Do not include the course number. In the box labeled Form write the appropriate test form letter A.Darken the appropriate circles below your ID number and Form designation.

Use a #2 pencil; machine grading may ignore faintly marked circles.

Mark your answers to the test questions in rows 112 of the op-scan sheet. You have 1 hour to complete

this part of the final exam. Your score on this part of the final exam will be the number of correct answers.

Please turn in your op-scan sheet and the question sheet at the end of this part of the final exam.

1. Ify(t) satisfies ty + y = t with y(1) = 1, then y(2) equals

(a)

5

4 (b) 2 (c) e2

(d)

1

e

2. Ify 2ty3 = 0 with y(2) = 1, then y satisfies

(a) y2 =1

9 2t2

(b) y2 = t3 t

(c) y3 =4

t2

(d) y = 12 2t2

3. Suppose A has eigenvectors

12

and

41

satisfying A

12

=

24

, A

41

=

82

.

Then a possible solution to Y = AY is

(a) Y =

e2t + 2e2t

2e2t + 4e2t

(b) Y =

5e2t

3e2t

(c) Y =

e2t + 4te2t

e2t + te2t

(d) Y =

e2t + 4e2t

2e2t + e2t

1


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4. The general solution of y + y = 0 is

(a) y = c1 + c2et + c3e

t + c4tet + c5t

2et

(b) y = c1 + c2t + c3 cos t + c4 sin t

(c) y = c1 + c2t + c3t2 + c4 cos t + c5 sin t

(d) y = c1 + c2t + c3t2 + c4e

t + c5tet

5. Consider the inhomogeneous differential equation y 3y + 2y = 11 + et

. If you try to

find a particular solution of the form yp = uet + ve2t, then u and v may satisfy:

(a) u = et

1 + et, v =

e2t

1 + et

(b) u = e2t

1 + et, v = e

t

1 + et

(c) u =1 + et

e3t, v =

1 + et

e3t

(d) u =1

(1 + et)e3t, v =

1

(1 + et)e3t

6. A particular solution of y 4y + 4y = te2t + t may be found of the following form(where the Ki have yet to be determined):

(a) yp = K1t2 + K2t + K3t

3e2t + K4t2e2t

(b) yp = K1t2 + K2t

2e2t + K3te2t

(c) yp = K1 + K2t + K3e2t + K4te

2t

(d) yp = K1t + K2te2t + K3e

2t

2


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7. The differential equation u + 2u u + 3u = 0 can be transformed to the first orderlinear system of the form X = AX, where the matrix A is given by

(a)

0 1 00 0 1

3 1 2

(b)

0 1 01 0 03 1 2

(c)

2 1 30 1 0

0 0 1

(d) 0 1 00 0 12 1 3

8. Which of the following differential equations are linear?

(A) y 1y

= 0

(B) (1 x2)yy = x

(C) y exy + (1 + x2)y = tan x

(D) y 2y + 3(1 y) = x2 + y

(a) Only (C)

(b) Only (A) and (C)

(c) Only (C) and (D)

(d) None of the differential equations

3


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9. The largest interval on which the initial value problem y =t

ywith y(2) = 1 has a

unique solution is the interval

(a) (,) (b) (0,) (c) (

3,) (d) (1, 3)

10. Let A =

1 91 1

. You are given that = 4 is an eigenvalue for A. Then a corre-

sponding eigenvector is

(a)

13

(b)

31

(c)

13

(d)

19

11. Suppose N grams of a radioactive material decays according to the usual decay lawN = kN, where k is the decay rate. If N1 = N(1) is the number of grams of undecayedmaterial after 1 time unit, and N10 = N(10) is the number of grams of undecayed materialafter 10 time units, then a formula for the decay rate k is

(a) k =ln 2

10

(b) k =ln 2

N1N10

(c) k = 9 + ln N10N1

(d) k = 19

lnN10

N1

12. Suppose you decide to approximate the solution of the initial value problem y =2y + 4t, y(0) = 1, using the Improved Euler Method with step size h = 1

2. Then your

approximate value for y(1) will be

(a) y(1) 3 (b) y(1) 5 (c) y(1) 192

(d) y(1) 474

4


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Math 2214 Fall 2001

Common Final Exam, Fall 2001

1. Which of the following differential equations is of second order and linear?

(a) y2y + ty + t2y1 = 0

(b) y + tsiny+y = tet

(c) y1 + (lnt)(y + 1)2t2 = 0

(d) (y1)y(y + 1) = 0

2. The general solution of the homogeneous differential equation y 2y +y = 0 isyh = c1e

x + c2xex. If a particular solution of the inhomogeneous differential equation

y2y+y = ex/x is sought of the form yp = uex +vxex and u is found to be u = x,

then v can be

(a) 1/x (b) x/e2x (c) lnx (d) 1

3. Suppose A is a 22 matrix with repeating eigenvalue = 5,5. Suppose u and v arevectors satisfying (A5I)u = 0 and (A5I)v =u. Then the general solution of thesystem differential equations Y = AY is:

(a) Y= c1e5tv+ c2e

5t(tv+u)

(b) Y= c1e5tv+ c2e

5t(tu+v)

(c) Y= c1e5tu+ c2e

5t(tv+u)

(d) Y= c1e5tu+ c2e

5t(tu+v)

4. Let yh denote the general solution of the homogeneous equation y +y2y = 0 and

let yp1 denote a particular solution to the inhomogeneous equation

y +y2y = etsin t.

Then the general solution of the inhomogeneous equation

y +y2y = etsint4t

is

(a) y = yh + 2t+ 1

(b) y = yh +yp1 + 2t+ 1

(c) y = yh +yp1 + 2t

(d) y = yh + t+12


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5. A solution of the initial value problem ty +y

t3= tan t, y(1) = 4 is guaranteed to

exist in the interval

(a) (,4) (b) (0,3) (c) (/2,3/2) (d) (0,/2)

6. Suppose you wish a numerical approximation to the solution of the initial value prob-

lem y = y+ 10t+ 10, y(0) = 5. If you use h = 0.2 and the Euler method to approxi-mate y(0.4), the result will be

(a) 8 (b) 12 (c) 17.2 (d) 24.4

7. Consider the differential equation y +ysin t= 0 with y(/2) = 1. Then y(0) is

(a) e (b) e (c) e1 (d) e1

8. The general solution to the differential equation y(4)y = 0 is

(a) c1 + c2t+ c3 sin t+ c4 cos t

(b) c1(1 + t) +c2 sint+ c3 cos t

(c) c1 + c2t+ c3et+c4e

t

(d) c1(1 + t) +c2et+ c3e

t

9. Determine the coefficients b and c such that etsin t and etcos t are solutions of the

differential equation y +by + cy = 0.

(a) b = 2, c = 2 (b) b = 2, c = 2 (c) b = 2, c = 2 (d) b = 2, c = 2

10. The eigenvalues of the matrix A =

1 3

2 2

are =1,4. The solution ofY = AY

satisfying Y(0) =

1

1

is

(a) e4t

e4t (b)

e4t

e4t (c)

14e4t+ 3

4et

1

3e4t

2

3et (d)

e4t

et


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11. The 22 matrix A =

3 5

1 1

has eigenvalues and eigenvectors = 1 + i,v =

5

2 i

and = 1 i, v =

52 + i

. Then a real-valued solution of the system of differential

equations Y = AY is

(a)

5etcos t

2etcos t+ etsin t

(b)

5et

2et

(c)

5etcos t

2etsin t

(d)

5et(cos t+ sint)2et(cos t sint)

12. Suppose an unusual decay law were given by dN/dt= N2, rather than the usuallinear decay law. If the initial amount was N(0) = N0, then what would be a formulafor the time t for which N(t) = N0/2?

(a) t=1

N0(b) t= ln 2 (c) t=

ln 2

N0(d) t=

N0

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