The following problems consider the “beats” that occur when the forcing term of a differential equation causes “slow" and “fast” amplitudes. Consider the general differential equation a y ″ + b y = cos ( ω t ) that governs undamped motion. Assume that b a ≠ ω . 139. Assuming the system starts from rest, show that the particular solution can be written as y = 2 a ( ω 0 2 − ω 2 ) sin ( ω 0 − ω t 2 ) sin ( ω 0 + ω t 2 ) .
The following problems consider the “beats” that occur when the forcing term of a differential equation causes “slow" and “fast” amplitudes. Consider the general differential equation a y ″ + b y = cos ( ω t ) that governs undamped motion. Assume that b a ≠ ω . 139. Assuming the system starts from rest, show that the particular solution can be written as y = 2 a ( ω 0 2 − ω 2 ) sin ( ω 0 − ω t 2 ) sin ( ω 0 + ω t 2 ) .
The following problems consider the “beats” that occur when the forcing term of a differential equation causes “slow" and “fast” amplitudes. Consider the general differential equation
a
y
″
+
b
y
=
cos
(
ω
t
)
that governs undamped motion. Assume that
b
a
≠
ω
.
139. Assuming the system starts from rest, show that the particular solution can be written as
y
=
2
a
(
ω
0
2
−
ω
2
)
sin
(
ω
0
−
ω
t
2
)
sin
(
ω
0
+
ω
t
2
)
.
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