1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

6 Chapter 0 Ordinary Differential Equations

0 <b/ 2 <ω:underdamped. Therootsarecomplexconjugatesα±iβwith
α=−b/2,β=

√

ω^2 −(b/ 2 )^2 .Thegeneralsolutionofthedifferentialequa-
tion is

u(t)=e−bt/^2

(

c 1 cos(βt)+c 2 sin(βt)

)

.

The mass oscillates, but approaches equilibrium astincreases.

b/ 2 =ω:critically damped. Therootsarebothequaltob/2. The general
solution of the differential equation is

u(t)=e−bt/^2 (c 1 +c 2 t).

The mass approaches equilibrium astincreases and may pass through equi-
librium (u(t)may change sign) at most once.

b/ 2 >ω:overdamped. Both roots of the characteristic equation are real,
say,m 1 andm 2 .Thegeneralsolutionofthedifferentialequationis

u(t)=c 1 em^1 t+c 2 em^2 t.

The mass approaches equilibrium astincreases, andu(t)may change sign at
most once. These cases are illustrated on the CD.

2. Cauchy–Euler equation

One of the few equations with variable coefficients that can be solved in com-
plete generality is the Cauchy–Euler equation:

t^2

d^2 u

dt^2 +kt

du

dt+pu=^0. (17)

The distinguishing feature of this equation is that the coefficient of thenth
derivative is thenth power oft, multiplied by a constant. The style of solution
for this equation is quite similar to the preceding: Assume that a solution has
the formu(t)=tm,andthenfindm. Substitutinguin this form into Eq. (17)
leads to

t^2 m(m− 1 )tm−^2 +ktmtm−^1 +ptm= 0 , or

m(m− 1 )+km+p= 0 (k,pare constants). (18)

This is the characteristic equation for Eq. (17), and the nature of its roots de-
termines the solution, as summarized in Table 2.
One important example of the Cauchy–Euler equation is

t^2 d

(^2) u
dt^2 +t
du
dt−λ
(^2) u= 0 , (19)