name
:
Mathematics 238 first test
Monday, July 7, 2008
please show your work to get full credit for each problem
1
. Is
y
=
e
4
x
a solution to the differential equation
y

5
y
+ 4
y
= 0
?
2
. Is
y
=
x
0
cos
t
2
dt
a solution of the initial value problem
dy
dx
= cos
x
2
and
y
(0) = 0 ?
3
. Given the differential equation
dy
dx
= (
x
2
sec 3
x
)
y
(
a
) Is the equation separable?
(
b
) Is the equation linear?
(
c
) Is the equation exact?
4
. Given the differential equation 3
x
2
y
2
+ 2
x
3
y
+ 12
y
2
dy
dx
= 0
(
a
) Is the equation separable?
(
b
) Is the equation linear?
(
c
) Is the equation exact?
5
. Solve the initial value problem
dy
dx
=
y
3
(2
x

5) and
y
(0) =
√
3
6
and state the solution’s domain.
6
. Solve the initial value problem
dy
dx
+3
y
= 6
x
and
y
(0) = 2
.
7
. Solve the differential equation
x
dy
dx

2
y
=
x
3
sin 4
x.
8
. Solve the initial value problem

2
x

3
y
3
+
1
x
dx
+ 3
x

2
y
2
+ 2
e
2
y
dy
= 0 and
y
(1) = 0
.
9
. Solve the differential equation
dy
dx
+4
y
= 2
e

2
x
√
y
using the substitution
w
=
√
y
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10
. Sketch the direction field & representative isoclines for
dy
dx
= 2+
x
y
11
. Suppose that
dP
dt
=
f
(
P
) where
f
is sketched below
(
a
) What are the equilibrium (or constant) solutions?
(
b
) Sketch a phase line for
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 Spring '09
 EWQFQW
 Derivative, Boundary value problem, dy

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