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1.3 Initial Conditions; InitialValue Problems
As we noted in the preceding section, we can obtain a particular solution of an
n
th order diﬀerential
equation simply by assigning speciFc values to the
n
constants in the general solution. However, in
typical applications of diﬀerential equations you will be asked to Fnd a solution of a given equation
that satisFes certain
preassigned
conditions.
Example 1.
±ind a solution of
y
±
=3
x
2

2
x
that passes through the point (1
,
3).
SOLUTION
In this case, we can Fnd the general solution by integrating:
y
=
±
(
3
x
2

2
x
)
dx
=
x
3

x
2
+
C.
The general solution is
y
=
x
3

x
2
+
C
.
To Fnd a solution that passes through the point (2
,
6), we set
x
= 2 and
y
= 6 in the general
solution and solve for
C
:
6=2
3

2
2
+
C
=8

4+
C
which implies
C
=2
.
Thus,
y
=
x
3

x
2
+ 2 is a solution of the diﬀerential equation that satisFes the given condition.
In fact, it is the only solution that satisFes the condition since the general solution represented all
solutions of the equation and the constant
C
was uniquely determined.
±
Example 2.
±ind a solution of
x
2
y
±±

2
xy
±
+2
y
=4
x
3
which passes through the point (1
,
4) with slope 2.
SOLUTION
As shown in Example 4 in the preceding section, the general solution of the diﬀerential
equation is
y
=
C
1
x
2
+
C
2
x
x
3
.
Setting
x
= 1 and
y
= 4 in the general solution yields the equation
C
1
+
C
2
+ 2 = 4
which implies
C
1
+
C
2
.
The second condition, slope 2 at
x
= 1, is a condition on
y
±
. We want
y
±
(1) = 2. We calculate
y
±
:
y
±
C
1
x
+
C
2
+6
x
2
,
and then set
x
= 1 and
y
±
= 2. This yields the equation
2
C
1
+
C
2
+ 6 = 2
which implies
2
C
1
+
C
2
=

4
.
Now we solve the two equations simultaneously:
11
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View Full DocumentC
1
+
C
2
=2
2
C
1
+
C
2
=

4
We get:
C
1
=

6
,C
2
= 8. A solution of the diﬀerential equation satisfying the two conditions is
y
(
x
)=

6
x
2
+8
x
+2
x
3
.
It will follow from our work in Chapter 3 that this is the only solution of the diﬀerential equation
that satisFes the given conditions.
±
INITIAL CONDITIONS
Conditions such as those imposed on the solutions in Examples 1 and
2 are called
initial conditions.
This term originated with applications where processes are usually
observed over time, starting with some initial state at time
t
=0
.
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 Spring '08
 Fonken

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