4—Differential Equations 73At timet= 0this is still zero even with the approximations. That’s comforting, but if it hadn’t
happened it’s not an insurmountable disaster. This is an approximation to the exact answer after all,
so it could happen that the initial conditions are obeyed only approximately. The exponential terms
have oscillations and damping, so the mass oscillates about its eventual equilibrium position and after
a long enough time the oscillations die out and you are left with the equilibrium solutionx=F 0 /k.
Look at point 4 above: For smallβtheβ^2 terms in Eq. (4.15) are small compared to theβterms
to which they are added or subtracted. The numerators of the terms witheαtare then proportional to
β. The denominator of the same terms has ak−bβin it. That means that asβ→ 0 , the numerator
of the homogeneous term approaches zero and its denominator doesn’t. The last terms, that came
from the inhomogeneous part, don’t have anyβin the numerator so they don’t vanish in this limit.
The approximate final result then comes solely from thexinh(t)term.
x(t)≈F 0
1
k
(
1 −e−βt
)
It doesn’t oscillate at all and just gradually moves from equilibrium to equilibrium as time goes on. It’s
what you get if you go back to the differential equation (4.12) and say that the acceleration and the
velocity are negligible.
m
d^2 x
dt^2
[≈0] =−kx−b
dx
dt
[≈0] +Fext(t) =⇒ x≈
1
k
Fext(t)
The spring force nearly balances the external force at all times; this is “quasi-static,” in which the
external force is turned on so slowly that it doesn’t cause any oscillations.
4.3 Series Solutions
A linear, second order differential equation can always be rearranged into the form
y′′+P(x)y′+Q(x)y=R(x) (4.17)
If at some pointx 0 the functionsP andQare well-behaved, if they have convergent power series
expansions aboutx 0 , then this point is called a “regular point” and you can expect good behavior of
the solutions there — at least ifRis also regular there.
I’ll look just at the case for which the inhomogeneous termR= 0. IfPorQhas a singularity
atx 0 , perhaps something such as 1 /(x−x 0 )or
√
x−x 0 , thenx 0 is called a “singular point” of the
differential equation.
Regular Singular Points
The most important special case of a singular point is the “regular singular point” for which the behaviors
ofPandQare not too bad. Specifically this requires that(x−x 0 )P(x)and(x−x 0 )^2 Q(x)have no
singularity atx 0. For example
y′′+
1
x
y′+
1
x^2
y= 0 and y′′+
1
x^2
y′+xy= 0
have singular points atx= 0, but the first one is a regular singular point and the second one is not.
The importance of a regular singular point is that there is a procedure guaranteed to find a solution near
a regular singular point (Frobenius series). For the more general singular point there is no guaranteed
procedure (though there are a few tricks* that sometimes work).
- The book by Bender and Orszag: “Advanced mathematical methods for scientists and engineers”
is a very readable source for this and many other topics.