Game Development Reference
In-Depth Information
Equating the coefficients of the sine and cosine terms from each side, we find
that
=
4
and
=
0
.
(13.56)
D
E
()
Thus, the function
p t
=
4sin
t
is a particular solution to Equation (13.51). The
complete solution is given by
()
x t
=
A t
cos 2
+
B t
sin 2
+
4 sin
t
(13.57)
or, equivalently,
()
(
)
x t
=
C
sin 2
t δ
+
+
4sin
t
,
(13.58)
where A , B , C , and δ are arbitrary constants.
13.2.3 Initial Conditions
In every solution to a second-order differential equation presented so far, there
have been two arbitrary constants. These constants allow for the specification of
certain initial conditions that dictate the values of
()
()
x t and
x
t
when
=
. Sup-
t
0
()
()
pose that the initial value of
x t is required to be x and the initial value of
x
t
()
is required to be v . Then the arbitrary constants appearing in the function
x t
can be determined by examining the following system of equations.
()
()
x
0
0
=
x
0
x
=
v
(13.59)
0
This is demonstrated in the following examples.
Example 13.5. Solve the differential equation
′′
()
()
()
x
t
5
x
t
+
6
x t
=
0
(13.60)
subject to the initial conditions
()
()
x
x
03
00
=
=
.
(13.61)