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School
California State University, Northridge *
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Course
280
Subject
Mathematics
Date
May 5, 2024
Type
Pages
38
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Homework 0.1
:
A)
Practice with the following integrals.
1.
1
2
ln
|
2
x
+ 1
|
+
C
2.
√
x
2
+ 1 +
C
3.
1
4
ln(
x
4
+ 1) +
C
4.
1
2
arctan(
x
2
) +
C
5.
x
+ 2
−
2 ln
|
x
+ 2
|
+
C
6.
1
2
ln
|
x
−
1
| −
1
2
ln
|
x
+ 1
|
+
C
7.
ln
|
x
−
2
| −
1
2
ln(
x
2
+ 1) + arctan(
x
) +
C
8.
−
1
x
+ ln
|
x
| −
ln
|
x
+ 1
|
+
C.
9.
1
3
x
2
e
3
x
−
2
9
xe
3
x
+
2
27
e
3
x
+
C.
10.
1
3
e
x
3
+
C.
11.
arcsin(
e
x
) +
C
12.
sin(ln(
x
)) +
C
13.
1
3
ln
|
sec(3
x
) + tan(3
x
)
|
+
C
14.
sec(
x
) +
C
15.
x
tan(
x
)
−
ln
|
sec(
x
)
|
+
C.
16.
x
ln(
x
)
−
x
+
C.
1
Homework 1.1
:
A)
For each ODE, find the order and classify as linear or nonlinear.
1.
1
-st order, linear.
2.
2
nd
order, nonlinear.
3.
5
-th order, nonlinear.
4.
3
-rd order, linear.
5.
2
-nd order, linear.
6.
1
-st order, nonlinear.
7.
3
-rd order, linear.
8.
1
-st order, linear.
9.
2
-nd order, nonlinear.
10.
2
-nd order, linear.
11.
1
-st order, nonlinear.
12.
3
-rd order, linear.
B)
Show that each function is a solution to the IVP. On what interval is the solution defined?
13.
I
= (
−∞
,
∞
)
.
14.
I
= (0
,
2)
.
15.
I
= (2
,
∞
)
.
16.
I
= (
−
π
2
,
π
2
)
.
17.
I
= (
−∞
,
∞
)
.
18.
I
= (0
,
∞
)
19.
I
= (
−∞
,
1)
.
20.
I
= (
π
2
,
3
π
2
)
.
21.
I
= (
−
π
4
,
π
4
)
.
22.
I
= (
−
3
π
4
,
π
4
)
.
23.
I
= (
−∞
,
3)
.
24.
I
= (2
,
∞
)
.
25.
I
= (
−∞
,
∞
)
.
26.
I
= (
−∞
,
0)
.
27.
I
= (
−
5
,
5)
.
28.
I
= (
−
2
,
∞
)
.
29.
I
= (
−
1
,
1)
.
30.
I
= (1
,
∞
)
2
Homework 1.2
:
A)
Solve for an explicit solution
y
(
x
)
.
1.
y
=
(
1
4
arctan
(
x
2
)
+
C
)
2
2.
y
= tan(
−
.
5
x
−
2
+
C
)
3.
y
= (tan(3
x
+
C
))
1
3
4.
y
=
−
1
arctan(
x
)+
C
.
5.
y
=
(
1
2
e
−
x
+
C
)
2
6.
y
=
√
(
x
2
+
C
)
2
+ 1
7.
y
= sin(2
√
x
+
C
)
8.
y
= 2
/
(ln
2
(
x
) +
C
)
9.
y
=
1
2
ln
(
e
x
2
+
C
)
.
B)
Find the explicit solutions
y
(
x
)
to the initial value problems.
10.
y
= 4
x
3
.
11.
y
= 3
e
5
x
−
2
.
12.
y
= (
x
2
+ 1)
2
−
1 =
x
4
+ 2
x
2
.
13.
y
=
2
1+ln(
x
2
+1)
.
14.
y
= arctan(ln(
x
) + 1)
.
15.
y
= sin
(
x
2
+
π
6
)
.
16.
y
= tan
(
1
−
1
x
)
.
17.
y
=
1
−
e
x
1+
e
x
.
C)
Find the explicit solution
y
(
x
)
by separation of variables. Simplify so that there are no logarithms
in your final answer.
18.
y
=
Cx
5
e
−
x
19.
y
=
Cx
3
(
x >
0)
20.
y
=
C/x
21.
y
=
1
2
+
C
x
2
22.
y
= 2 +
Cx
3
23.
y
= 3 +
C
(1 +
.
2
x
)
5
24.
y
=
C
√
x
2
+ 1
25.
y
=
(
1 +
Ce
3
x
)
1
3
26.
y
=
C
x
e
2
x
2
27.
y
=
Cx
3
e
−
x
28.
y
= 3 +
Cx
2
e
x
29.
y
=
Ce
x
x
+1
30.
y
=
Ce
x
(
x
−
2)
3
31.
y
=
C
sec
3
(
x
)
32.
y
= arcsin(
Ce
−
x
2
)
33.
y
=
3
2
e
3
x
+1
.
34.
y
=
6
e
2
x
3
e
2
x
−
1
.
35.
y
=
1
−
e
2
x
e
2
x
+1
.
3
D)
Find the solutions to the IVP’s in implicit
form (Do not solve for
y
!).
36.
y
sin(
y
) + cos(
y
) =
1
4
x
2
−
5
.
37.
1
3
y
3
+
y
=
x
2
−
x
+ 10
.
38.
y
sin(
y
) + cos(
y
) =
x
+ 1
.
39.
ye
−
y
+
e
−
y
=
1
x
+ 2
.
40.
y
ln(
y
)
−
y
=
−
3 cos(
x
) + 2
.
41.
2 ln(
y
)
−
7
y
=
−
1
3
x
3
+ 2
.
42.
2 ln(
y
) + 3
y
−
1
=
x
2
+ 2
.
43.
y
−
ln(
y
+ 1) =
1
2
x
2
−
2
.
44.
ye
y
−
e
y
= 2
e
2
x
−
3
.
45.
2 ln(
y
−
2)
−
ln(
y
−
1) =
x
−
ln(2)
.
46.
ln(
y
) + arctan(
y
) =
x
+
π
4
.
E)
Write the explicit solution to the IVP using a
definite
integral.
47.
y
=
(
1
2
∫
x
3
cos
(
1
t
)
dt
+ 8
)
2
3
.
48.
y
=
∫
x
1
e
t
t
2
dt
+ 5
.
49.
y
=
∫
x
π
sin(
t
)
t
dt
+ 1
.
50.
y
=
(
3
∫
x
0
cos(
t
2
)
dt
+ 8
)
1
3
.
51.
y
= exp
(
∫
x
0
1
t
3
+1
dt
+ 1
)
.
52.
y
=
(
4
−
∫
x
e
1
ln(
t
)
dt
)
−
1
.
53.
y
=
(
1
2
∫
x
8
sin(
t
2
)
dt
+ 3
)
2
.
54.
y
=
√
∫
x
1
√
1 +
t
3
dt
+ 9
.
55.
y
= ln
(
1 +
∫
x
1
e
t
2
dt
)
.
4
Homework 1.3:
A)
Use the method of integrating factors to find the general solution.
1.
y
=
−
1
2
x
2
+
Ce
x
2
x
2
.
2.
y
=
1
2
e
−
3
x
+
Ce
−
5
x
3.
y
=
−
x
−
1 +
Ce
x
4.
y
=
xe
3
x
+
Ce
3
x
5.
y
=
x
+
Cx
−
2
6.
y
=
1
2
x
ln
2
(
x
) +
Cx
7.
y
= 2
x
3
+
Cx
5
8.
y
=
−
x
2
+
Cx
3
.
9.
y
=
e
x
3
3
x
3
+
C
x
3
.
10.
y
=
1
3
x
2
+
Cx
2
e
−
3
x
.
11.
y
=
1
x
−
1
x
2
+
Ce
−
x
x
2
.
12.
y
=
sin(
x
)
x
+
cos(
x
)
x
2
+
C
x
2
.
13.
y
= 1 +
C
(
x
+ 2)
−
3
.
14.
y
=
1
x
−
1
+
Ce
−
x
x
−
1
.
15.
y
=
1
2
x
3
+
Ce
−
x
2
x
3
.
16.
y
= 1 +
C
√
x
2
+1
.
17.
y
=
1
1
−
x
+
C
(
x
+1
1
−
x
)
.
18.
y
= sin(
x
) +
C
cos(
x
)
.
19.
y
=
x
−
cos(
x
)
sec(
x
)+tan(
x
)
+
C
sec
(
x
)+tan(
x
)
.
B)
Use the method of integrating factors to solve the IVP’s.
20.
y
=
−
1
2
x
+
1
4
x
3
.
21.
y
=
e
3
x
+ 2
e
x
.
22.
y
= 2
x
−
1 + 2
e
−
2
x
.
23.
y
=
1
x
−
cos(
x
2
)
x
.
24.
y
= 2
−
4
√
x
.
25.
y
= 1 + 2
e
−
x
2
.
26.
y
=
1
8
x
3
−
1
8
x
−
5
.
27.
y
=
x
3
−
x
2
.
28.
y
=
x
2
−
1
2
x
2
cos(2
x
)
.
29.
y
=
sin(
x
)
x
2
+
cos(
x
)
x
3
+
1
x
3
.
30.
y
= 1 +
8
(
x
+1)
3
.
31.
y
= 2
x
−
3
−
(
x
−
1) ln(
x
−
1)
.
32.
y
=
x
−
1 +
2
2
x
−
1
.
33.
y
=
1
2
e
x
−
2
x
2
e
x
.
34.
y
= 3
x
−
3
xe
−
2
x
+2
.
35.
y
=
1
3
x
3
+1
x
2
−
1
.
36.
y
= 2
√
x
2
+ 9
−
1
.
37.
y
= 1
−
2
e
cos(
x
)
38.
y
=
x
cos(
x
)
−
π
4
cos(
x
)
C)
Find the solution
y
(
x
)
to the initial value problem in terms of a
definite integral
.
39.
y
=
x
3
∫
x
2
t
−
4
sin(
t
)
dt
+ 3
x
3
.
40.
y
=
e
x
2
∫
x
1
e
−
t
2
dt
+ 2
e
x
2
41.
y
=
e
−
sin(
x
)
∫
x
π
t
2
e
sin(
t
)
dt
+ 3
e
−
sin(
x
)
42.
y
=
x
∫
x
3
e
2
t
t
2
dt
+ 2
x
43.
y
=
1
x
2
∫
x
3
t
sec(
t
)
dt
−
9
x
2
44.
y
=
1
√
x
∫
x
4
1
√
t
e
t
2
dt
+
2
√
x
45.
y
=
1
√
x
4
+8
∫
x
−
1
1
√
t
4
+8
dt
+
3
√
x
4
+8
5
Homework 1.4
:
A)
Sketch the solutions to the initial value problems. Use the direction fields from the pictures below.
1.
a)
b)
c)
2.
a)
b)
c)
3.
a)
b)
c)
4.
a)
b)
c)
B)
Sketch the general solutions.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
C)
Sketch the solutions to the initial value problems. (Your answer should be a
single function
. You
can use the general solution to find the answer, but the general solution should
not
be a part of your
answer!)
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
6
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