Game Development Reference
In-Depth Information
2
a
′′
()
()
()
xt axt
+
+
xt
=
0
.
(13.23)
4
It is a simple task to verify that the function
()
(
)
xt
=
te
at
(13.24)
is a solution to Equation (13.23), so the general solution to Equation (13.12)
when 1
rr
=
is given by
2
()
x t
=
e
rt
+
te
rt
,
(13.25)
where we have set
rr r
==
.
1
2
, then the roots of the auxiliary equation are complex. The solu-
tion given by Equation (13.19) is still correct, but it requires the use of complex
arithmetic. We can express the solution entirely in terms of real-valued functions
by using the formula
If
ab
2
−<
4
0
αβi
+
α
(
)
e
=
e
cos
β i β
+
sin
(13.26)
(see Appendix A, Section A.4). Assuming that a and b are real numbers, the roots
r and r of the auxiliary equation are complex conjugates, so we may write
r α i
r α i
=+
=−
1
,
(13.27)
2
where
a
α
=−
2
1
β
=
4
ba
2
.
(13.28)
2
The solution given by Equation (13.19) can now be written as
(
)
(
)
()
αβit
+
αβit
xt
=
Ae
+
Be
αt
(
)
αt
(
)
=
Ae
cos
βti βt
+
sin
+
e βti βt
cos
sin
αt
[
(
)
(
)
]
=
eAB βt
+
cos
+
ABi βt
sin
.
(13.29)
This solution can be expressed using two real constants C and C by setting