Math 251
October
12, 2005
First Exam
NAME:
Section #:
There are 9 questions on this exam. Many of them have multiple parts. The point value of each question
is indicated either at the beginning of each question or at the beginning of each part where there are
multiple part
Show all your work
. Partial credit may be given.
The use of calculators, books, or notes is not permitted on this exam.
Please turn off your cell phone before starting this exam.
Time limit 1 hour and 15 minutes.
Question
Score
1
22pt
2
10pt
3
10pt
4
8pt
5
12pt
6
10pt
7
10pt
8
8pt
9
10pt
Total
100pt
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1.
a. 2pt
Consider the following differential equation
y
=
y
+ 2
t
. Without solving it, determine the slope of
the tangent line to the solution at the point (1
,
2).
b. 2pt
Find the Wronskian
W
(
y
1
, y
2
) of the functions
y
1
= sin
t
and
y
2
= cos
t
c. 2pt
Suppose
y
1
and
y
2
are two solutions of the ODE
y
+ (sin
t
)
y
+
y
= 0. and suppose that
their Wronskian by
W
(
y
1
, y
2
)(
t
) is 2 at
t
= 0. Find
W
(
y
1
, y
2
)(
t
) for any
t
.
For the initial value problems in parts
d.
through
g.
state whether or not one of our two existence and uniqueness
theorems for first order ODE’s guarantees a unique solution. If the answer is yes and the theorem provides an interval
of existence, then state what the interval is
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 Spring '08
 CHEZHONGYUAN
 Math, Differential Equations, Equations, Partial Differential Equations, Ode, Constant of integration

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