4—Differential Equations 88- Find anyonesolution to the full inhomogeneous equation. Note that for step one you have to have
all the arbitrary constants present; for step two you do not. [xinh(t)] - Add the results of steps one and two. [xhom(t) +xinh(t)]
I’ve already done step one. To carry out the next step I’ll start with a particular case of the forcing function.
IfFext(t)is simple enough, you should be able toguessthe answer to step two. If it’s a constant, then a constant
will work forx. If it’s a sine or cosine, then you can guess that a sine or cosine or a combination of the two should
work. If it’s an exponential, then guess an exponential — remember that the derivative of an exponential is an
exponential. If it’s the sum of two terms, such as a constant and an exponential, it’s easy to verify that you add
the results that you get for the two cases separately. If the forcing function is too complicated for you to guess a
solution then there’s a general method using Green’s functions that I’ll get to later.
Choose a specific example
Fext(t) =F 0
[
1 −e−βt]
(12)
This starts at zero and builds up to a final value ofF 0. It does it slowly or quickly depending onβ.
F 0t
Start with the first term,F 0 , for external force in Eq. ( 11 ). Tryx(t) =Cand plug into that equation to
find
kC=F 0
This is simple and determinesC.
Next, use the second term as the forcing function,−F 0 e−βt. Guess a solutionx(t) =C′e−βtand plug in.
The exponential cancels, leaving
mC′β^2 −bC′β+kC′=−F 0 or C′=−F 0
mβ^2 −bβ+kThe total solution for the inhomogeneous part of the equation is then the sum of these two expressions.
xinh(t) =F 0(
1
k−
1
mβ^2 −bβ+ke−βt