Chapter 4
4.
The corresponding homogeneous equation,
y
+
y
= 0, has the associated auxiliary
equation
r
2
+
r
=
r
(
r
+ 1) = 0.
This gives
r
= 0,

1, and a general solution to the
homogeneous equation is
y
h
(
t
) =
c
1
+
c
2
e

t
. Combining this solution with the particular
solution,
y
p
(
t
) =
t
, we find that a general solution is given by
y
(
t
) =
y
p
(
t
) +
y
h
(
t
) =
t
+
c
1
+
c
2
e

t
.
6.
The corresponding auxiliary equation,
r
2
+ 5
r
+ 6 = 0, has the roots
r
=

3,

2.
Therefore, a general solution to the corresponding homogeneous equation has the form
y
h
(
x
) =
c
1
e

2
x
+
c
2
e

3
x
. By the superposition principle, a general solution to the original
nonhomogeneous equation is
y
(
x
) =
y
p
(
x
) +
y
h
(
x
) =
e
x
+
x
2
+
c
1
e

2
x
+
c
2
e

3
x
.
8.
First, we rewrite the equation in standard form, that is,
y

2
y
= 2 tan
3
x .
The corresponding homogeneous equation,
y

2
y
= 0, has the associated auxiliary
equation
r
2

2 = 0.
Thus
r
=
±
√
2, and a general solution to the homogeneous
equation is
y
h
(
x
) =
c
1
e
√
2
x
+
c
2
e

√
2
x
.
Combining this with the particular solution,
y
p
(
x
) = tan
x
, we find that a general solution
is given by
y
(
x
) =
y
p
(
x
) +
y
h
(
x
) = tan
x
+
c
1
e
√
2
x
+
c
2
e

√
2
x
.
10.
We can write the nonhomogeneous term as a difference
(
e
t
+
t
)
2
=
e
2
t
+ 2
te
t
+
t
2
=
g
1
(
t
) +
g
2
(
t
) +
g
3
(
t
)
.
The functions
g
1
(
t
),
g
2
(
t
), and
g
3
(
t
) have a form suitable for the method of undetermined
coefficients.
Therefore, we can apply this method to find particular solutions
y
p,
1
(
t
),
y
p,
2
(
t
), and
y
p,
3
(
t
) to
y

y
+
y
=
g
k
(
t
)
,
k
= 1
,
2
,
3
,
respectively. Then, by the superposition principle,
y
p
(
t
) =
y
p,
1
(
t
) +
y
p,
2
(
t
) +
y
p,
3
(
t
) is a
particular solution to the given equation.
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 Spring '08
 MAZMANI
 Equations, Constant of integration, general solution

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