Download Spring 2004 Math 2214 Common Final Exam and more Exams Differential Equations in PDF only on Docsity!
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 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 1-11 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.
- 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)
- The general solution of
y ¢¢¢+ 2 y ¢ ¢+ y ¢ = 0
is
(a)
y ( t ) = C 1
t + C 2
e
t
t e
(b)
y ( t ) = C 1
+ C
2
e
t
t e
t
(c)
y ( t ) = C 1
t + C 2
e
t
t e
t
(d) y ( t ) = C 1
+ C
2
e
t e
- Consider the initial value problem
y^ ¢= 2 t y
2
If we use Euler’s method with step size 1 to calculate
y ( 1 ), we obtain
(a) -8 (b) 0 (c) -4 (d) 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/
- 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 salt
in the tank at time t. Then
Q ( t ) is a solution of the initial value problem:
(a)
Q ¢ = -
Q
, Q (0) = 10
(b)
Q ¢ = -
Q
, Q (0) = 30
(c)
Q^ ¢ = 2 -
Q
, Q (0) = 10
(d) Q^ ¢ = 2 -
Q
, Q (0) = 10
- The differential equation
sin( t ) y ¢ + e
t
y = 1 is
(a) nonlinear (b) homogeneous (c) separable (d) linear
- The general solution of
x 1
( t )
x 2
( t )
Ê
Ë
Á
Á
Ê
Ë
Á
x 1
( t )
x 2
( t )
Ê
Ë
Á
is:
(a)
x 1
( t )
x 2
( t )
Ê
Ë
Á
= c 1
e
2 t
Ê
Ë
Á
e
Ê
Ë
Á
(b)
x 1
( t )
x 2
( t )
Ê
Ë
Á
= c 1
e
2 t
Ê
Ë
Á
e
2 t
Ê
Ë
Á
Ê
Ë
Á
È
Î
Í
(c)
x 1
( t )
x 2
( t )
Ê
Ë
Á
= c 1
e
2 t
Ê
Ë
Á
e
2 t
t
Ê
Ë
Á
Ê
Ë
Á
È
Î
Í
(d)
x 1
( t )
x 2
( t )
Ê
Ë
Á
˜ =^ c 1
Ê
Ë
Á
˜ +^ c 2
cos( 2 t )
Ê
Ë
Á
- The differential equation
y ¢¢+ y ¢= 5 t + 3 e
t
has the general solution:
(a)
y ( t ) = C 1
+ C
2
e
t
2
t
(b)
y ( t ) = C 1
+ C
2
e
e
t
(c)
y ( t ) = C 1
+ C
2
e
t
2
e
t
(d)
y ( t ) = C 1
+ C
2
e
t
2
t
- Consider the differential equation
t
2
y ¢¢+ t y ¢ - y = t
2
.
The corresponding homogeneous equation has the solutions
y 1
( t ) = t and
y 2
( t ) = t
.
The general solution of the given nonhomogeneous differential equation is
(a)
y ( t ) = c 1
t + c 2
t
(b)
y ( t ) = c 1
t + c 2
t
t
2
(c)
y ( t ) = c 1
t + c 2
t
2
(d) y ( t ) = c 1
t + c 2
t
t
2
