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)

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