Exercise Set 9.1
373
11.
µ
=
e
dx
=
e
x
,
e
x
y
=
e
x
cos(
e
x
)
dx
= sin(
e
x
) +
C
,
y
=
e
−
x
sin(
e
x
) +
Ce
−
x
12.
dy
dx
+ 2
y
=
1
2
,
µ
=
e
2
dx
=
e
2
x
,
e
2
x
y
=
1
2
e
2
x
dx
=
1
4
e
2
x
+
C
,
y
=
1
4
+
Ce
−
2
x
13.
dy
dx
+
x
x
2
+ 1
y
= 0
, µ
=
e
(
x/
(
x
2
+1))
dx
=
e
1
2
ln(
x
2
+1)
=
x
2
+ 1
,
d
dx
y
x
2
+ 1
= 0
, y
x
2
+ 1 =
C, y
=
C
√
x
2
+ 1
14.
dy
dx
+
y
=
1
1 +
e
x
,
µ
=
e
dx
=
e
x
,
e
x
y
=
e
x
1 +
e
x
dx
= ln(1 +
e
x
) +
C
,
y
=
e
−
x
ln(1 +
e
x
) +
Ce
−
x
15.
1
y
dy
=
1
x
dx
, ln

y

= ln

x

+
C
1
, ln
y
x
=
C
1
,
y
x
=
±
e
C
1
=
C
,
y
=
Cx
including
C
= 0 by inspection
16.
dy
1 +
y
2
=
x
2
dx,
tan
−
1
y
=
1
3
x
3
+
C, y
= tan
1
3
x
3
+
C
17.
dy
1 +
y
=
−
x
√
1 +
x
2
dx,
ln

1 +
y

=
−
1 +
x
2
+
C
1
,
1 +
y
=
±
e
−
√
1+
x
2
e
C
1
=
Ce
−
√
1+
x
2
,
y
=
Ce
−
√
1+
x
2
−
1
, C
= 0
18.
y dy
=
x
3
dx
1 +
x
4
,
y
2
2
=
1
4
ln(1 +
x
4
) +
C
1
,
2
y
2
= ln(1 +
x
4
) +
C, y
=
±
[ln(1 +
x
4
) +
C
]
/
2
19.
1
y
+
y
dy
=
e
x
dx,
ln

y

+
y
2
/
2 =
e
x
+
C
; by inspection,
y
= 0 is also a solution
20.
dy
y
=
−
x dx,
ln

y

=
−
x
2
/
2 +
C
1
, y
=
±
e
C
1
e
−
x
2
/
2
=
Ce
−
x
2
/
2
, including
C
= 0 by inspection
21.
e
y
dy
=
sin
x
cos
2
x
dx
= sec
x
tan
x dx
,
e
y
= sec
x
+
C
,
y
= ln(sec
x
+
C
)
22.
dy
1 +
y
2
= (1 +
x
)
dx,
tan
−
1
y
=
x
+
x
2
2
+
C, y
= tan(
x
+
x
2
/
2 +
C
)
23.
dy
y
2
−
y
=
dx
sin
x
,
−
1
y
+
1
y
−
1
dy
=
csc
x dx,
ln
y
−
1
y
= ln

csc
x
−
cot
x

+
C
1
,
y
−
1
y
=
±
e
C
1
(csc
x
−
cot
x
) =
C
(csc
x
−
cot
x
)
, y
=
1
1
−
C
(csc
x
−
cot
x
)
, C
= 0;
by inspection,
y
= 0 is also a solution, as is
y
= 1.
24.
1
tan
y
dy
=
3
sec
x
dx
,
cos
y
sin
y
dy
= 3 cos
x dx
, ln

sin
y

= 3 sin
x
+
C
1
,
sin
y
=
±
e
3 sin
x
+
C
1
=
±
e
C
1
e
3 sin
x
=
Ce
3 sin
x
, C
= 0,
y
= sin
−
1
(
Ce
3 sin
x
)
, as is
y
= 0 by inspection