1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Chapter 0 Ordinary Differential Equations 5

Figure 1 Mass–spring–damper system.

test shows them to be independent. Therefore, we may equally well write

u(t)=c 1 ′cosh(λt)+c′ 2 sinh(λt)

as the general solution of Eq. (14), wherec′ 1 andc 2 ′are arbitrary constants.

Example: Mass–Spring–Damper System.
The displacement of a mass in a mass–spring–damper system (Fig. 1) is de-
scribed by the initial value problem

d^2 u

dt^2

+bdu

dt

+ω^2 u= 0 ,

u( 0 )=u 0

du
dt(^0 )=v^0.
The equation is derived from Newton’s second law. Coefficientsbandω^2
are proportional to characteristic constants of the damper and the spring, re-
spectively. The characteristic equation of the differential equation is

m^2 +bm+ω^2 = 0 ,

with roots

−b±

√

b^2 − 4 ω^2

2

=−b

2

±

√(

b

2

) 2

−ω^2.

The nature of the solution, and therefore the motion of the mass, is determined
by the relation betweenb/2andω.

b= 0 :undamped. The roots are±iωandthegeneralsolutionofthediffer-
ential equation is

u(t)=c 1 cos(ωt)+c 2 sin(ωt).

The mass oscillates forever.