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Initial Value - Introduction Differential Equations - Exam, Exams of Differential Equations

This is the Exam of Introduction Differential Equations and its key important points are: Initial Value, Solution is Guaranteed, Constant Matrix, Possible Solution, Mixture, Stirred and Drains, Same Rate, Approximate, Constant Matrix, Linear Differential Equation

Typology: Exams

2012/2013

Uploaded on 02/14/2013

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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 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 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. Consider the initial value problem tan( ) ln( ), (2) 1ytyty
+
==. Find the largest
interval in which a solution is guaranteed to exist.
(a) ( (b) ( (c) (1, 2) 0,1) / 2,3 / 2)
π
π
(d) (1,3 / 2)
π
2. Let A be a constant matrix such that
(2 2)×48
12
A

=


and 12
24
A

=


. One
possible solution of the system A
=
yy
is:
(a) (b) (c)
22
2
4
2
tt
t
ee
ee

+

+

2tt
22
22
212
43
tt
t
ee
ee

+

+

2
2
2
t
t
e
e
(d)
22
22
28
42
tt
tt
ete
ete



3. Let solve the initial value problem
( )
yt 2 , (0) 1yy t y
=
=. What is the value ? (2)y
(a)
y= (b) y (c) y
(2) 3 (2) 5=2
(2) e
=
(d) ye
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 the
tank 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 tank
contains 180 lb of salt.
(a) te (b) te (c) t
180/ 200
=200/ 180
=200 ln 9
=
(d) tln(180/ 200)
=
pf3

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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 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 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. Consider the initial value problem y ′^ + tan( ) t y = ln( ) , t y (2) = 1. Find the largest interval in which a solution is guaranteed to exist.

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

  1. Let A be a (2 × 2)constant matrix such that

A

and

A ^ ^ =  

. One possible solution of the system y ′^ = A y is:

(a) (b) (c)

2 2 2

t t t

e e e e

− −

 2 t^  t

2 2 2 2

t t t

e e e e

− −

2 2 2

t t

e e

(d)

2 2 2 2

t t t t

e te e te

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

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

  1. 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 the tank 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 tank contains 180 lb of salt.

(a) t = e 180 / 200 (b) t = e 200 /180 (c) t = 200 ln 9 (d) t =ln(180 / 200)

  1. Let y t ( ) solve the initial value problem ty ′^ + 4 y = 6 t^2 , y (1)= 1

e

. What is y (2)?

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

  1. Suppose you decide to approximate the solution of ty ′^ + y^2 = 4 t^2 , y (1) = 4 using Euler’s method with step size h = 1/ 2. Your approximate value for y (2) is:

(a) − 5 (b) 6 − 7 (c) 6

− (d) 1 3

  1. Let A be a (2 × 2)constant matrix with real entries such that^1 (3 2 )^1 1 1

A ^ +^^ i^ ^ = + i  + i      .

The solution of , (0) 5 2

′ = A = ^ 

y y y is:

(a) e^3 (b) e 5cos 2 5sin 2 2 cos 2 3sin 2

t t^ t t t

3 5cos 2^ sin 2 2 cos 2 3sin 2

t t^ t t t

(c) e^3 (d) e 5cos 2 5sin 2 2 cos 2 sin 2

t t^ t t t

^ +

3 5cos 2^ sin 2 2 cos 2 sin 2

t t^ t t t

  1. The solution of the following initial value problem

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

(a) 1 + 2 te −^3 tet^ (b) 2 − ete^3 t (c) 1 − t (d) (1/ 6) + (1/12) e −^3 t^ +(3 / 4) et

  1. One solution of the 2nd^ order linear differential equation

t y^2 '' + 2 ty ' − 2 y = 0, ( t ≠ 0 ),

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

(a) y = c t 1 + c t 2 −^3 (b) y = c t 1 + c t 2 −^1 (c) y = c t 1 + 2 c t 2 (d) y = c t 1 + c t 2 −^2