Math 333 Logarithms Worksheet With Answers Page 2
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13The initial conditions imply that
0 = y(0) = 1 + c
1
so that c
=
1, and
1
1 = y (0) = 18
c
+ 3c
1
2
so that c
=
6.
2
The complete solution is
y(t) = cos 3t + 6 sin 3t
e (cos 3t + 6 sin 3t).
The first two terms are the steady-state solution. The terms multiplied by e
are the transient solution.
(b) Compare the amplitude and phase angle of the steady-state solution with those
of the function 37 cos 3t.
Solution: The steady-state solution above can be written as
37 cos(3t
ϕ)
1
6
where cos ϕ =
and sin ϕ =
. The angle ϕ with these trig values is
37
37
approximately 1.4056 radians, or about 80.5 degrees. The steady-state response
has a smaller amplitude than the forcing function (by a factor of
37) and lags
about 80.5 degrees behind the forcing function.
Here is a picture showing the driving function (with the larger amplitude) and
the response function:
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