
224 Classical Mechanics
© 2010 Taylor & Francis Group, LLC
Putting this into Equation 7.58 yields
At
Ft
e
()
()
=
−ω
0
. (7.61)
Integration gives the solution of Equation 7.58:
ye
Ft
m
ety
it it
t
=
′
′
+
−
′
∫
0
0
()
()d
(7.62)
where y(0) is the constant of integration, the value of y at the time t = 0. Equation 7.62 is the required
general solution of a rst-order differential equation, Equation 7.58. From Equation 7.61, we see that
the function x(t) is given by the imaginary part of Equation 7.62, divided by ω
0
:
xt
yt
()
Im
=
ω
0
. (7.63)
Force F(t) in Equation 7.62 or 7.63 must be written in real form.
The energy of the system cannot be conserved as the sy ...