Mathematical Tools for Physics

4—Differential Equations 91

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. ( 13 ) are small compared to theβterms to which
they are added or subtracted. The numerators of the terms witheαtare then proportional toβ. The denominator
of the same terms has ak−bβin it. That means that it doesn’t go to zero asβgoes to zero. 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 just 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 ( 11 ) 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-equilibrium.”

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) (15)

If at some pointx 0 the functionsPandQare 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 only at the case for which the inhomogeneous termR= 0. IfP orQhas 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 ofP